Gaussian statistics for palaeomagnetic vectors
Gaussian statistics for palaeomagnetic vectors
复制标题
古地磁矢量的高斯统计
DOI：
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发表时间：
2003
期刊：
影响因子：
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通讯作者：
C. Constable
中科院分区：
文献类型：
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作者：
J. Love;C. Constable
总结
与治疗的目的是统计的古地磁方向和强度联合和一致，我们代表的平均值和方差的古地磁矢量，在一个特定的网站和一个特定的极性，由概率密度函数在笛卡尔三空间的正交磁场分量组成的一个单一的（单峰）非零平均值，球对称（各向同性）高斯函数。对于古地磁数据的混合极性，我们认为一个双峰分布组成的一对这样的对称高斯函数，具有相等的，但相反的，平均值和等方差。对于高斯和双高斯分布，并在球面三空间的强度，倾斜度，和decermination，我们获得的边缘密度函数，累积分布，以及每个球坐标（包括相对于对称轴的角度分布）的期望值和方差的解析表达式。强度和离轴角的数学表达式是封闭形式的，并且特别易于管理，其中强度分布是瑞利-莱斯的。在小的相对矢量色散的限制下，高斯（双高斯）方向分布接近于Fisher（Bingham）分布，并且强度分布接近于正态分布。在大的相对矢量色散的相反极限下，方向分布接近球形均匀分布，强度分布接近麦克斯韦分布。我们量化的偏见，估计的性质的向量场产生的使用简单的算术平均值，如估计的强度或倾斜的平均向量，或这些量的方差。随着统计框架的发展，并使用最大似然法，它给出了无偏估计的大数据数量的限制，我们演示了如何制定的逆问题，以及如何估计的平均值和方差的磁矢量场，即使当数据组成的混合组合的方向和强度。我们研究古地磁长期变化数据从夏威夷和留尼汪岛，虽然这两个网站是在几乎相反的纬度，我们发现显着差异的平均矢量和差异的地方矢量方差，与夏威夷的数据是特别各向异性。这些意见是不一致的平均场的描述是一个简单的地心轴向偶极子和长期变化是统计对称的反射通过赤道平面。最后，我们的分析古地磁采集数据从1960年基拉韦厄流在夏威夷和全新世Xitle流在墨西哥，是一致的广泛持有的怀疑，方向数据比强度数据更准确。
SUMMARY
With the aim of treating the statistics of palaeomagnetic directions and intensities jointly and consistently, we represent the mean and the variance of palaeomagnetic vectors, at a particular site and of a particular polarity, by a probability density function in a Cartesian three-space of orthogonal magnetic-field components consisting of a single (unimodal) non-zero mean, spherically-symmetrical (isotropic) Gaussian function. For palaeomagnetic data of mixed polarities, we consider a bimodal distribution consisting of a pair of such symmetrical Gaussian functions, with equal, but opposite, means and equal variances. For both the Gaussian and bi-Gaussian distributions, and in the spherical three-space of intensity, inclination, and declination, we obtain analytical expressions for the marginal density functions, the cumulative distributions, and the expected values and variances for each spherical coordinate (including the angle with respect to the axis of symmetry of the distributions). The mathematical expressions for the intensity and off-axis angle are closed-form and especially manageable, with the intensity distribution being Rayleigh–Rician. In the limit of small relative vectorial dispersion, the Gaussian (bi-Gaussian) directional distribution approaches a Fisher (Bingham) distribution and the intensity distribution approaches a normal distribution. In the opposite limit of large relative vectorial dispersion, the directional distributions approach a spherically-uniform distribution and the intensity distribution approaches a Maxwell distribution. We quantify biases in estimating the properties of the vector field resulting from the use of simple arithmetic averages, such as estimates of the intensity or the inclination of the mean vector, or the variances of these quantities. With the statistical framework developed here and using the maximum-likelihood method, which gives unbiased estimates in the limit of large data numbers, we demonstrate how to formulate the inverse problem, and how to estimate the mean and variance of the magnetic vector field, even when the data consist of mixed combinations of directions and intensities. We examine palaeomagnetic secular-variation data from Hawaii and Reunion, and although these two sites are on almost opposite latitudes, we find significant differences in the mean vector and differences in the local vectorial variances, with the Hawaiian data being particularly anisotropic. These observations are inconsistent with a description of the mean field as being a simple geocentric axial dipole and with secular variation being statistically symmetrical with respect to reflection through the equatorial plane. Finally, our analysis of palaeomagnetic acquisition data from the 1960 Kilauea flow in Hawaii and the Holocene Xitle flow in Mexico, is consistent with the widely held suspicion that directional data are more accurate than intensity data.