Isotropic reproducing kernels for the inner of a sphere or spherical shell and their use as density covariance functions

Isotropic reproducing kernels for a sphere or spherical shell are derived as weighted product sums o fL ₂ orthonormal base functions. For the sphere these functions are products of the surface spherical harmonics and the Jacobi polynomials of degree (0, 2). Reproducing kernels for a sphere are consistent with the covariance function of the outer anomalous gravity potential of the Earth. These reproducing kernels may be used for gravity field modeling which include density (anomaly) data as observations or which aims at predicting such quantities using optimal estimation methods, that is for solving the inverse gravimetric problem.     

Transformation of global spherical harmonic models of the gravity field to a local adjusted spherical cap harmonic model

The new global gravity models represented by global spherical harmonics like EGM2008 require a high degree and order in their coefficients to resolve the gravity field in local areas; therefore, there are interests to represent the regional or local field by less parameters and to develop a parameter transformation from the global model to a local kind of spherical harmonic model. The authors use local spherical cap harmonics for the regional gravity potential representation related to a local pole and a local spherical coordinate system. This allows to model regional gravity potential with less parameters and less memory requirements in computation and storage. From different kinds of representations of spherical cap harmonics, we have selected the so-called adjusted spherical cap harmonics (ASCH). This is the most appropriate for the presented mathematical model of deriving its coefficients from global gravity models. In that way, the global gravity models can fully be exploited and mapped to regional ASCH, in particular with respect to the computation of regional geoid models with improved solution.     

Spherical cap harmonics revisited and their relationship to ordinary spherical harmonics

The “global” representation of the geomagnetic field in terms of ordinary spherical harmonics (SHs) and its corresponding set {g,h} of coefficients has been studied extensively, but the “local” representation in terms of spherical cap harmonics (SCHs) and its corresponding set {G,H} of coefficients is not yet well understood. This paper clarifies some of the main properties of the SCHs and their proper use along with their relationship with the SHs. In particular, it shows that for the spherical cap part of a global field specified by spherical harmonics there is a strict relation between the ordinary Legendre functions of the global representation and the fractional functions of the local expansion; hence we can express the set of coefficients {G,H} in terms of the set {g,h}. Finally, some attention will be given to the role of the leading (n = 0, m = 0) term of the SCH expansion.     

Upper mantle lateral inhomogeneity estimated using global MVS

We processed synchronous observatory MVS data at periods from 1.2 to 6 days to estimate lateral conductivity variations in the upper mantle. The inverse problem for depth-dependent conductivity distribution was solved using correlation of the measured surface components of the geomagnetic field. We applied spherical harmonic expansion of the complex amplitudes of time harmonics and multiple averaging of the amplitudes obtained for different SH sets. The chosen approximation approach was tested by comparing the EM field simulated for a smoothly inhomogeneous Earth with the exact forward solution. The SHA-analyzed observatory data for fifteen magnetic storms were used to map global apparent conductivity variations.     

Optimization fitting of the eccentric dipole models to the observed geomagnetic field

We have obtained formulas to calculate the field components of an eccentric dipole (ED) with an arbitrarily directed moment from specified coordinates of the dipole center and from its moment vector. With these formulas, the model dipole position was fitted to the observed geomagnetic field, and the approximation accuracy was estimated according to the standard deviation from IGRF along the X, Y, and Z components; the fitting procedure was checked in several tests.We computed the ED parameters (position and moment orientation) and harmonic coefficients of the ED field for 50 years using surface IGRF components, discovered some changes, and compared them with those according to the Schmidt eccentric dipole model. It was found that the nondipolar contributions to the geomagnetic field increase as the dipole field decays with time, and the dipole center drifts northand eastward away from the Earth's gravity center.The main contribution of the dipole part in the Gaussian spherical harmonic expansion of the geomagnetic field turned out to come from the terms with n of 1 to 5 rather than from two first terms, and the contribution of higher harmonics increases with time. Therefore, the Schmidt ED approximation based on the first eight Gauss coefficients (n ≤ 2) must have only relative significance indicating changes of the parameters. On the other hand, world magnetic anomalies (WMA) show up already in two first Gauss terms rather than since the third term as it has been commonly assumed.     

基于地磁参数和子午线收敛角计算的井斜方位角校正方法

井斜方位角的校正对于定向井轨迹控制、中靶分析有非常重要的意义。通过分析大地坐标与高斯克吕格投影坐标之间的关系,根据地磁场球谐函数模型,地磁参数和子午线收敛角的计算模型,提出了实现方式和求解步骤,并编制了相应的计算软件。通过实例计算与分析,表明地磁参数和子午线收敛角的准确计算对于定向井轨迹控制的重要性。     

Closed Form Solutions of the Two-Dimensional Turning Bands Equation

The turning bands method generates realizations of isotropic Gaussian random fields by means of appropriately summed line processes. For two-dimensional simulations the relation between the isotropic correlation function of the random field and the correlation function to be simulated along the lines is given by an integral equation of Abel type. We present closed form solutions of this integral equation for almost all two-dimensional correlation models encountered in practice and discuss their numerical implementation. As an additional benefit, our tables and illustrations serve as a concise guide to correlation models useful in geostatistics.     

Manifestation of Singlets of the Basic Spheroidal Mode of the Earth 0S2 in Geomagnetic Variations

The results of instrumental observations have shown that the spectra of geomagnetic variations exhibit a quasi-harmonic component, the frequency of which is close to the basic spheroidal mode of the Earth ₀S₂. In periods 15 days long after three large earthquakes and in the absence of strong magnetic disturbances, the fine structure of this mode in geomagnetic variations is identified in the form of singlets. The frequencies of these singlets are similar to the theoretical values calculated using an ideal elastic isotropic model of the Earth of spherical symmetric form without its rotation.

     

APPLICATION OF MAGNETOMETRIC DATA IN THE STUDY OF DEEP CRUSTAL STRUCTURE IN PLATFORM PROVINCES

Results of regional geophysical work in the U.S.S.R. allow a measure of correlation between the known structural features of the crust and those of the geomagnetic field. The “normal” magnetic field used for identifying magnetic anomalies is obtained by graphic smoothing of isolines of total intensity T. Detailed analysis of data obtained in aeromagnetic and gravity surveys throughout the U. S. S. R. leads to the conclusion that the cause of the observed regional magnetic and gravity anomalies lies in the same associated geologic objects located at the same level - that of a magnetoactive interval of the crystalline envelope of the earth. The conclusions that can be drawn from existing data are: 1) the existence of “continental” anomalies of the geomagnetic field cannot be explained as an effect of the crust; 2) the basaltic layer does not affect the distribution of the geomagnetic field; 3) the anomalies in the basic (normal) geomagnetic field are caused by the presence of thermomagnetic geologic bodies in the granitic layer; accordingly, it is theoretically possible to determine the thickness of the continental layer from magnetometric data. A table showing correlative depths of the basaltic layer, its thickness, and anomalous magnetic field, as well as magnetic profiles is included.--C. E. Sears.     

Local earth gravity/potential modeling using ASCH

In Geodesy, the heights of points are normally orthometric heights measured above the geoid (an equipotential surface created by the earth masses and rotation which approximately coincides with the mean sea level) or the normal heights. It is necessary to transform the GNSS/GPS measured ellipsoidal heights (h) to classical physical heights (orthometric H/Normal H). The total gravity potential of the earth (W) is the summation of two components; gravitational potential (V) by earth masses and the centrifugal potential (Ω). The centrifugal potential is directly calculated, while the gravitational potential (V) needs to be modeled globally or locally using given measurements. The global models of the earth gravitational potential/gravity models (or so-called geoid models) are mostly given using spherical harmonics (SH). A modified approach of SH was defined to fit the use of SH for regional gravity/potential modeling called spherical cap harmonics (SCH). Due to the numerical difficulties of SCH, a simplified approach of SCH is selected to be used for a combined modeling of the earth potential using a variety of observations. This approach is called the Adjusted Spherical Cap harmonics.     

The large-scale solar magnetic field and 11-year activity cycles

Magnetic Hα synoptic maps of the Sun for 1915–1999 are analyzed and the intensities of spherical harmonics of the large-scale solar magnetic field computed. The possibility of using these Hα maps as a database for investigations of long-term variations of solar activity is demonstrated. As an example, the magnetic-field polarity distribution for the Hα maps and the analogous polarity distribution for the magnetographic maps of the Stanford observatory for 1975–1999 are compared. An activity index A(t) is introduced for the large-scale magnetic field, which is the sum of the magnetic-moment intensities for the dipole and octupole components. The 11-year cycle of the large-scale solar magnetic field leads the 11-year sunspot cycle by, on average, 5.5 years. It is concluded that the observed weak large-scale solar magnetic field is not the product of the decay of strong active-region fields. Based on the new data, the level of the current (23rd) solar-activity cycle and some aspects of solar-cycle theory are discussed.     

Lateral lithospheric conductivity patterns,from European synchronous observatory data

Much information on the regional lithospheric structure may come from MTS data acquired by synchronous 2D arrays and processed with regard to the nonlocal response of a laterally inhomogeneous subsurface. We suggest to invert the nonlocal MT responses applying correlation of all surface horizontal and vertical components of the geomagnetic field recorded simultaneously at all stations. The inversion algorithm has been applied to 2004–2005 European observatory data of diurnal S_q variations for first five harmonics and yielded lateral conductivity patterns for different periods. The maps show spatial correlation between conductivity maxima and lithospheric thickness minima and, specifically, highlight the contours of the Pannonian basin, where lithosphere is as thin as ～50 km, from seismic data.     

Study of the Interaction between Biological Cells of Different Shapes and Sizes and Electromagnetic Fields Produced by Natural Phenomena

We carry out a study of the interaction between electromagnetic fields produced innature and cells with no structure but with different shapes and sizes, considered asapproximations of those observed in the human body. Magnetic fields produced byvarious atmospheric and geomagnetic phenomena are compared with the fieldsgenerated by the cells themselves. Working with frequencies up to 10⁵ Hz, we found that in most cases lightning is the natural phenomena capable of producingfield intensities comparable to those generated by the cells. As a first approximationwe assume that a cell might be affected only if a field of intensity comparable to thecell's field is present. Oscillating fields of less intensity and their harmonics can involveresonant interactions with the cells, however this situation is not considered here. Under the approximation of our study we found that the size and the shape are parameters of paramount importance: lightning represents a natural hazard only for the largest cells, while the quadrangular cells are not affected by any of the natural phenomena considered here.     

地球重力场的球面小波分析研究

高精度高分辨率地球重力场模型是地球科学、空间科学以及海洋科学发展的需要。目前的全球重力场模型的阶数已达 36 0阶、50km× 50km的分辨率。调和展开分析是当前高精度全球重力场模型的基础 ,由于全球外调和展开存在只有频域的局部性而缺乏空间域的局部性的缺陷 ,使得地球重力场模型向高频逼近时不可能在外调和框架下有效地进行。利用球面小波分析和多尺度分析理论可克服该缺陷。通过发展球面小波理论 ,来对地球重力场进行球面小波展开 ,获得地球重力场的多尺度结构 ,建立局部或全球重力场的多尺度模型 ,揭示地球重力场的多尺度结构与场源密度分布之间的响应 ,恢复重力场的空间域与频率域的精细结构 ,提高重力场的精度和分辨率 ,以推进地球重力场在相关学科的基础、战略作用。     

球谐、频谱分析在雅鲁藏布江大拐弯地区地磁研究中的应用

本文利用科学院地球物理研究所的绝对地磁图(1970,1980)数据,采用球谱分析原理提取区域场,以频谱分析原理计算居理面深度,对西藏雅江大拐弯地区25万Km²范围的地磁场进行了数据处理。结合该区的地质、地热、地震等方面的资料,综合分析了该区地磁场及磁性基底的变化规律,推断了该区内深大断裂构造线的位置、走向、倾向等。这些结果对了解该区深部地质情况提供了参考信息。     

Features of discrete VLF emissions observed at Gulmarg,India during the magnetic storm of 6–7 March, 1986

During the analysis of archived VLF data from Indian low latitude ground stations, some discrete VLF emissions recorded at the low latitude ground station Gulmarg (geomagnetic latitude 24°26′N; geomagnetic longitude 147°09′E, L = 1.28) during moderate magnetic storm activity (Σ K _P ⁻ = 32, K _P index varies from 4 to 6 during the observation period) on 6/7 March, 1986 are presented in this paper. The dynamic spectra of these discrete VLF emissions were observed along with tweeks and its harmonics, which is interesting and complex to explain. In most of the events the harmonic frequency of tweeks correlates with the starting frequency of harmonics of discrete emissions. In order to explain the observed features of discrete VLF emissions, we propose cyclotron resonance interaction between whistler mode wave and energetic electrons of inner radiation belt as possible generation mechanism. An attempt is also made to determine parallel energy, anisotropy and wave growth relevant to the generation process of VLF emissions.     

External field characterization using CHAMP satellite data for induction studies

Knowledge of external inducing source field morphology is essential for precise estimation of electromagnetic (EM) induction response. A better characterization of the external source field of magnetospheric origin can be achieved by decomposing it into outer and inner magnetospheric contributions, which are best represented in Geocentric Solar Magnetospheric (GSM) and Solar Magnetic (SM) reference frames, respectively. Thus we propose a spherical harmonic (SH) model to estimate the outer magnetospheric contribution, following the iterative reweighted least squares approach, using the vector magnetic data of the CHAMP satellite. The data covers almost a complete solar cycle from July 2001 to September 2010, spanning 54,474 orbits. The SH model, developed using orbit-averaged vector magnetic data, reveals the existence of a stable outer magnetospheric contribution of about 7.39 nT. This stable field was removed from the CHAMP data after transforming to SM frame. The residual field in the SM frame acts as a primary source for induction in the Earth. The analysis of this time-series using wavelet transformation showed a dominant 27-day periodicity of the geomagnetic field. Therefore, we calculated the inductive EM C-response function in a least squares sense considering the 27-day period variation as the inducing signal. From the estimated C-response, we have determined that the global depth to the perfect substitute conductor is about 1132 km and its conductivity is around 1.05 S/m.     

Long and short term relationships between solar wind velocity and geomagnetic field at low latitudes

Solar wind velocity control of low latitude geomagnetic field both on long and short term basis is studied. It is shown that semiannual averages of the low latitude field is inversely related to solar wind velocity and that there is a dominant local time dependence of the relationship. Strongest correlation are confined to the local afternoon hours. It is also shown that for a duration when the solar wind velocity exhibits significant recurrent pattern the low latitude geomagnetic field also depicts strong solar synodic rotation periodicity of 27 days with significant coherence with velocity. The low latitude field on a short term basis is influenced by variable solar wind velocity with a delay of about 1–2 days. During the period of systematic recurrent pattern in solar wind velocity even the quiet-time night field at equatorial and low latitudes show a strong dependence on velocity indicative of the solar wind control of the quiet-time proton belt encompassing the earth.     

地磁总强度梯度的计算与分析

不仅要知道地磁场的分布规律,而且要知道地磁场梯度的分布规律。本上根据地磁场的高斯理论和地磁场的球谐模型,计算出中国及其邻近地区地磁总强度的水平梯度(南北方向和东西方向)和垂直梯度,绘制出相应的等值线图,并分析了总强度梯度的分布规律。地磁总强度的水平梯度和垂直梯度,主要随纬度变化,与经度关系不大。值得注意的是在我国的中心地区,地磁总强度的水平梯度值最大。     

Use of Surface Waves in Statistical Correlations of Shear Wave Velocity and Penetration Resistance of Chennai Soils

Shear wave velocity (V _s) is one of the most important input parameter to represent the stiffness of the soil layers. It is preferable to measure V _s by in situ wave propagation tests, however it is often not economically feasible to perform the tests at all locations. Hence, a reliable correlation between V _s and standard penetration test blow counts (SPT-N) would be a considerable advantage. This paper presents the development of empirical correlations between V _s and SPT-N value for different categories of soil in Chennai city characterized by complex variation of soil conditions. The extensive shear wave velocity measurement was carried out using Multichannel Analysis of Surface Waves (MASW) technique at the sites where the SPT-N values are available. The bender element test is performed to compare the field MASW test results for clayey soils. The correlations between shear wave velocity and SPT-N with and without energy corrections were developed for three categories of soil: all soils, sand and clay. The proposed correlations between uncorrected and energy corrected SPT-N were compared with regression equations proposed by various other investigators and found that the developed correlations exhibit good prediction performance. The proposed uncorrected and energy corrected SPT-N relationships show a slight variation in the statistical analysis indicating that both the uncorrected and energy corrected correlations can predict shear wave velocity with equal accuracy. It is also found that the soil type has a little effect on these correlations below SPT-N value of about 10.