• Bulletin of the Korean Mathematical Society
• /
• v.50 no.5
• /
• pp.1451-1469
• /
• 2013

We present various new sharp assertions on multipliers in mixed norm, weighted Hardy and new Lizorkin-Triebel spaces of harmonic functions in higher dimension. Some results are new even in onedimensional case.

• Journal of the Korean Society of Surveying, Geodesy, Photogrammetry and Cartography
• /
• v.24 no.2
• /
• pp.183-191
• /
• 2006

The accuracy of the upward continuation is assessed through the gravity modeling using an ultra-high degree spherical harmonic expansion. The difficulties in the numerical calculation of Legendre function with ultra-high degree, underflow and/or overflow, is successfully resolved in 128 bit calculation scheme. Using the generated Legendre function, the gravity anomaly with spatial resolution of $1'{\times}1'$ on the geoid is calculated. The generated gravity anomaly is degraded and extracted with various noise levels and data intervals, then upward continuation is applied to each data sets. The comparison between the upward continued gravity disturbances and the directly calculated from the spherical harmonics showed that the accuracy on the direct method was significantly better than that of Poisson method. In addition, it is verified that the denser and less noised gravity data on the geoid generates better gravity disturbance vectors at an altitude. Especially, it is found that the gravity noise level less than 5mGal, and the data interval less than 2arcmin is necessary for next generation precision INS navigation which requires the accuracy of 5mGal or better at an altitude.

• Journal of the Korean earth science society
• /
• v.37 no.2
• /
• pp.107-116
• /
• 2016

Stability of the barotropic Rossby-Haurwitz wave is investigated using the numerical models on the global domain. The Rossby-Haurwitz wave under investigation is composed of the basic zonal flow of super-rotation and a finite amplitude spherical harmonic wave. The Rossby-Haurwitz wave is given as either steady or unsteady wave by adjusting the strength of the super-rotating zonal flow. Stability as well as the growth rate of the wave in the numerical simulation is determined by comparing the perturbation amplitude at two different time stages. Unstable modes of the Rossby-Haurwitz wave exhibited a horizontal structure composing of various zonal-wavenumber components. The vorticity perturbation for some modes showed a discontinuity around the area of weak flow, which was found robust regardless of the horizontal resolution of the model. Fourier finite element model was shown to generate the unstable mode in earlier stage of the time integration due to less accuracy compared to the spherical harmonic spectral model. Taking the overall accuracy of the models into consideration, the time by which the unstable mode begin to dominate over the spherical harmonic wave was estimated.

• Journal of the Institute of Electronics Engineers of Korea CI
• /
• v.43 no.2 s.308
• /
• pp.72-80
• /
• 2006

In this paper, we propose a new descriptor for 3D model retrieval based on shape information. The proposed method consists of two steps including ray casting method and spherical harmonic function, considering geometric properties of model. In the ray casting method, an adaptive sampling is performed for external shape information. By increasing shape information included in the descriptor, we improve the discriminating power of the proposed descriptor. The coefficients of spherical harmonic function are adaptively calculated, considering geometric frequency characteristics. This makes the descriptor more compact and concise without decreasing the retrieval performance. By combining two methods, we achieve more improved retrieval results.

• Economic and Environmental Geology
• /
• v.49 no.4
• /
• pp.269-280
• /
• 2016

I presented three vector crustal magnetic anomaly components and six gradients by using spherical Slepian functions over the cap area of $20^{\circ}$ of radius centered on the South Pole. The Swarm mission, launched by European Space Agency(ESA) in November of 2013, was planned to put three satellites into the low-Earth orbits, two in parallel in East-West direction and one in cross-over of the higher altitude. This orbit configuration will make the gradient measurements possible in North-South direction, vertical direction, as well as E-W direction. The gravity satellites, such as GRACE and GOCE, have already implemented their gradient measurements for recovering the accurate gravity of the Earth and its temporal variation due to mass changes on the subsurface. However, the magnetic gradients have little been applied since Swarm launched. A localized magnetic modeling method is useful in taking an account for a region where data availability was limited or of interest was special. In particular, computation to get the localized solutions is much more efficient and it has an advantage of presenting high frequency anomaly features with numbers of solutions fewer than the global ones. Besides, these localized basis functions that were done by a linear transformation of the spherical harmonic functions, are orthogonal so that they can be used for power spectrum analysis by transforming the global spherical harmonic coefficients. I anticipate in scientific and technical progress in the localized modeling with the gradient measurements from Swarm and here will do discussion on the results of the localized solution to represent the three vector and six gradient anomalies over the Antarctic area from the synthetic data derived from a global solution of the spherical harmonics for the crustal magnetic anomalies of Swarm measurements.

• Economic and Environmental Geology
• /
• v.12 no.2
• /
• pp.95-104
• /
• 1979

The position of any point on the earth's surface can be. represented in the spherical coordinates by surface spherical harmonics. Since geomagnetic field is a function of position on the earth, it can be also expressed by spherical harmonic analysis as spherical harmonics of trigonometric series of $a_m({\theta})$ cos $m{\phi}$ and $b_m({\theta})$ sin $m{\phi}$. Coefficients of surface spherical harmonics, $a_m({\theta})$ and $b_m({\theta})$, can be drawn from the components of the geomagnetic field, declination and inclination, and vice versa. In this paper, components of geomagnetic field, declination and inclination in the Korean peninsula are obtained by spherical harmonic analysis using the Gauss coefficients calculated from the world-wide magnetic charts of 1960. These components correspond to the values of normal geomagnetic field having no disturbances of subsurface mass, structure, and so on. The vertical and total components offer the zero level for the interpretation of geomagnetic data obtained by magnetic measurement in the Korean peninsula. Using this zero level, magnetic anomaly map is obtained from the data of airborne magnetic. prospecting carried out during 1958 to 1960. The conclusions of this study are as follows; (1) The intensity of horizontal component of normal geomagnetic field in Korean peninsula ranges from $2{\times}10^4$ gammas to $2.45{\times}10^4$ gammas. It decreases about 500 with the increment of $1^{\circ}$ in latitude. Along the same. latitude, it increases 250 gammas with the increment of $1^{\circ}$ in longitude. (2) Intensity of vertical component ranges from $3.85{\times}10^4$ gammas to $5.15{\times}10^4$ gammas. It increases. about 1000 gammas with the increment of $1^{\circ}$ in latitude. Along the same latitude, it decreases. 150~240 gammas with the increment of $1^{\circ}$ in longitude. Decreasing rate is considerably larger in higher latitude than in lower latitude. (3) Total intensity ranges from $4.55{\times}10^4$ gammas to $5.15{\times}10^4$ gammas. It increases 600~700 gammas with the increament of $1^{\circ}$ in latitude. Along the same latitude, it decreases 10~90 gammas. with the increment of $1^{\circ}$ in longitude. Decreasing rate is considerably larger in higher latitude as the case of vertical component. (4) The declination ranges from $-3.8^{\circ}$ to $-11.5^{\circ}$. It increases $0.6^{\circ}$ with the increment of $1^{\circ}$ in latitude. Along the same latutude, it increases $0.6^{\circ}$ with the increment of l O in longitude. Unlike the cases of vertical and total component, the rate of change is considerably larger in lower latitude than in higher latitude. (5) The inclination ranges from $57.8^{\circ}$ to $66.8^{\circ}$. It increases about $1^{\circ}$ with 'the increment of $1^{\circ}$ in latitude Along the same latitude, it dereases $0.4^{\circ}$ with the increment of $1^{\circ}$ in longitude. (6) The Boundaries of 5 anomaly zones classified on the basis of the trend and shape of anomaly curves correspond to the geologic boundaries. (7) The trend of anomaly curves in each anomaly zone is closely related to the geologic structure developed in the corresponding zone. That is, it relates to the fault in the 3rd zone, the intrusion. of granite in the 1st and 5th zones, and mountains in the 2nd and 4th zones.

• Bulletin of the Korean Chemical Society
• /
• v.4 no.1
• /
• pp.17-25
• /
• 1983

The hyperfine integrals for 4d orbitals have been evaluated adopting a general method which is applicable to a general vector, R, pointing arbitrary direction in space. The operator and the spherical harmonic part of 4d orbitals are expressed in terms of R and r$_{N}$ and the exponential part, r$^{2}$exp(-2${\beta}$r), of 4d orbitals is also translated as a function of R and r$_{N}$ and then integration is performed. The radial integrals for 4d orbitals are tabulated in analytical forms. The hyperfine integrals for 4d orbitals are also represented in analytical forms, using the specific formulas of radial series which we found.

• Journal of Astronomy and Space Sciences
• /
• v.26 no.3
• /
• pp.287-294
• /
• 2009

In helioseismology it is normally assumed that p-mode oscillations are excited in a statistically independent fashion. Unfortunately, however, this issue is not clearly settled down in that two experiments exist, which apparently look in discrepancy. That is, Appourchaux et al. (2000) looked at bin-to-bin correlation and found no evidence that the assumption is invalid. On the other hand, Roth (2001) reported that p-mode pairs with nearby frequencies tend to be anti-correlated, possibly by a mode-coupling effect. This work is motivated by an idea that one may test if there exists an excess of anticorrelated power variations of pairs of solar p-modes. We have analyzed a 72-day MDI spherical-harmonic time series to examine temporal variations of p-mode power and their correlation. The power variation is computed by a running-window method after the previous study by Roth (2001), and then distribution function of power correlation between mode pairs is produced. We have confirmed Roth's result that there is an excess of anti-correlated p-mode pairs with nearby frequencies. On the other hand, the amount of excess was somewhat smaller than the previous study. Moreover, the distribution function does not exhibit significant change when we paired modes with non-nearby frequencies, implying that the excess is not due to mode coupling. We conclude that the origin of this excess of anticorrelations may not be a solar physical process, by pointing out the possibility of statistical bias playing the central role in producing the excess.

• Journal of Astronomy and Space Sciences
• /
• v.32 no.3
• /
• pp.247-256
• /
• 2015

In this work, an efficient method with which to evaluate the high-degree-and-order gravitational harmonics of the non-sphericity of a central body is described and applied to state predictions of a lunar orbiter. Unlike the work of Song et al. (2010), which used a conventional computation method to process gravitational harmonic coefficients, the current work adapted a well-known recursion formula that directly uses fully normalized associated Legendre functions to compute the acceleration due to the non-sphericity of the moon. With the formulated algorithms, the states of a lunar orbiting satellite are predicted and its performance is validated in comparisons with solutions obtained from STK/Astrogator. The predicted differences in the orbital states between STK/Astrogator and the current work all remain at a position of less than 1 m with velocity accuracy levels of less than 1 mm/s, even with different orbital inclinations. The effectiveness of the current algorithm, in terms of both the computation time and the degree of accuracy degradation, is also shown in comparisons with results obtained from earlier work. It is expected that the proposed algorithm can be used as a foundation for the development of an operational flight dynamics subsystem for future lunar exploration missions by Korea. It can also be used to analyze missions which require very close operations to the moon.

• Journal of the Korean earth science society
• /
• v.35 no.5
• /
• pp.333-341
• /
• 2014

A new set of basis functions was constructed using the Rossby-Haurwitz waves, which are the eigenfunctions of nondivergent barotropic vorticity equations on the sphere. The basis functions were designed to be non-separable, that is, not factored into functions of either the longitude or the latitude. Due to this property, the nodal lines of the functions are aligned neither along with the meridian nor the parallel. The basis functions can be categorized into groups of which members have the same degree or the total wavenumber-like index on the sphere. The orthonormality of the basis functions were found to be close to the machine roundoffs, giving the error of $O(10^{-15})$ or $O(10^{-16})$ for double-precision computation (64 bit arithmetic). It was demonstrated through time-stepping procedure that the basis functions were also the eigenfunctions of the non-divergent barotropic vorticity equations. The projection of the basis functions was carried out onto the low-resolution geopotential field of Gaussian bell, and compared with the theory. The same projections were performed for the observed atmospheric-geopotential height field of 500 hPa surface to demonstrate decomposition into the fields that contain disturbance of certain range of horizontal scales. The usefulness of the new basis functions was thus addressed for application to the eigenmode analysis of the atmospheric motions on the global domain.