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1.
ANALYSIS OF DISPERSION ON A SPHERE 总被引：1，自引：0，他引：1
G. S. Watson 《Geophysical Journal International》1956,7(S4):153-159
i
The probability density on a sphere, exp ( K cos θ'), where θ' is the angle between the polar and observation vectors, has recently been suggested by Fisher for the analysis of palaeo-magnetic data. In this paper, a series of approximate tests of significance is given for hypotheses about the precision constant K and the polar vector 相似文献
The probability density on a sphere, exp ( K cos θ'), where θ' is the angle between the polar and observation vectors, has recently been suggested by Fisher for the analysis of palaeo-magnetic data. In this paper, a series of approximate tests of significance is given for hypotheses about the precision constant K and the polar vector 相似文献
2.
Summary. Advantages of using the mode in analysis of palaeomagnetic vectors are discussed, and a computer technique is described for contouring and precisely locating the modes of vector distributions that may be highly skewed. In contrast to conventional determinations of the mode, unit vectors from a given data set are treated not as discrete points, but as identical Fisherian probability density functions defined (at an angle θ from the unit vector) by: p = exp [ sk (cos θ– 1)], where kis the estimate of the Fisherian concentration parameter, and s is an arbitrarily assigned 'smoothing parameter'. A grid, representing the cumulative probability distribution of the total sample of vectors, is contoured to provide a graphical display of the distribution around the most probable value, the mode. By repeatedly contouring the same sample of vectors with successively larger values of s, and by treating the mode as a vector with length given by the total probability value at the mode, 'progressive modal diagrams' can be constructed, to aid in determining the stable position of the mode of skewed distributions. In addition, a new statistic β95 is suggested as an error estimator for the mode. The statistic β95 is derived from the largest subset of the total sample that has a mean identical with the mode of the total sample; this statistic is defined as the Fisherian half-angle of the cone of 95 per cent confidence for the mean of this subset. 相似文献
3.
Summary. A simple normal approximation is given for the joint probability density function of the polar coordinates (θ, φ) of a random vector following the Fisher distribution with arbitrary mean direction (θ0 , φ0 ). Two important applications of this approximation are discussed, and conditions for the validity of the approximation are investigated and summarized. 相似文献
4.
STANLEY N.DEMING JOHN A.PALASOTA Department of Chemistry University of Houston Houston TX - U.S.A.AND JOHN M.NOCERINO United States Environmental Protection Agency Environmental Monitoring Systems Laboratory Las Vegas NV - U.S.A. Author to whom correspondence should be addressed. 《地理学报(英文版)》1993,(5)
The geometric properties of three common object-preprocessing transformations(constant sum,orclosure;constant length,or normalization;and maximum value,or ratioing)are investigated.Anargument is made for using absolute values in the constant sum and maximum value transformations.In general,each transformation distorts the shape and dimensionality of patterns in the data:transformed data lie on(C-l)-dimensional surfaces in the original C-dimensional space.A data set thathas been closed by one of these transformations can be reopened if a vector containing the constant sums,constant lengths or maximum values of the original objects was retained.Transformed data sets may befreely interconverted among these three transformations without the loss of information. 相似文献
5.
Summary. This paper describes the statistical techniques available to the experimenter in palaeomagnetic work. The theory of these methods is based on an assumed probability distribution of errors. It is shown that the mathematical requirements of this distribution are obeyed by the observations from rock samples which are known to possess a stable magnetization; observations on rocks with unstable magnetization however do not conform to it. A theoretical derivation is given for this probability distribution.
The problem of estimating the mean direction of magnetization of a geological formation has in recent years become a matter of the greatest geophysical interest since it is from such estimates that the position of the pole of the Earth in past geological ages is determined. This problem is largely one of the judicious choice of samples and a procedure is suggested whereby such estimates may be achieved with the greatest sample economy. 相似文献
The problem of estimating the mean direction of magnetization of a geological formation has in recent years become a matter of the greatest geophysical interest since it is from such estimates that the position of the pole of the Earth in past geological ages is determined. This problem is largely one of the judicious choice of samples and a procedure is suggested whereby such estimates may be achieved with the greatest sample economy. 相似文献
6.
P. L. McFadden 《Geophysical Journal International》1980,60(3):391-396
Summary. The angle, θ p , in a Fisher distribution which will be exceeded with probability p is shown to be given by The solution does not involve any estimation of the population precision and as such is the small sample solution. Fisher derived the analogue of 'Student's' test for mean directions and the test derived here is the equivalent test for individual observations. 相似文献
7.
EDMUND R.MALINOWSKI 《地理学报(英文版)》1988,(1)
Fisher variance ratio tests are developed for determining(1)the number of statistically significantabstract factors responsible for a data matrix and(2)the significance of target vectors projected into theabstract factor space.F-tests,developed from the viewpoint of vector distributions,are applied tovarious data sets taken from the chemical literature. 相似文献
8.
We present a neural network approach to invert surface wave data for a global model of crustal thickness with corresponding uncertainties. We model the a posteriori probability distribution of Moho depth as a mixture of Gaussians and let the various parameters of the mixture model be given by the outputs of a conventional neural network. We show how such a network can be trained on a set of random samples to give a continuous approximation to the inverse relation in a compact and computationally efficient form. The trained networks are applied to real data consisting of fundamental mode Love and Rayleigh phase and group velocity maps. For each inversion, performed on a 2°× 2° grid globally, we obtain the a posteriori probability distribution of Moho depth. From this distribution any desired statistic such as mean and variance can be computed. The obtained results are compared with current knowledge of crustal structure. Generally our results are in good agreement with other crustal models. However in certain regions such as central Africa and the backarc of the Rocky Mountains we observe a thinner crust than the other models propose. We also see evidence for thickening of oceanic crust with increasing age. In applications, characterized by repeated inversion of similar data, the neural network approach proves to be very efficient. In particular, the speed of the individual inversions and the possibility of modelling the whole a posteriori probability distribution of the model parameters make neural networks a promising tool in seismic tomography. 相似文献
9.
Tatiana Quarta Maurizio Fedi & Angelo De Santis 《Geophysical Journal International》2000,140(2):311-323
An analysis of the field scaling power spectrum yields useful information about the source distribution, but it is uncertain whether deterministic, random, fractal or mixed approaches have to be used for the interpretation. To this end, the scaling properties of potential field spectra are analysed for a number of different source models of geological interest. Besides the models of Naidu (purely random sources) and Spector and Grant (gross block statistical ensembles) we consider other types of density and magnetization distributions with spectral exponents in the fractal range, such as a single homogeneous body with a random white source distribution. Spectral slopes in the fractal range are obtained.
We also study the effects of important natural sources, such as salt domes and sedimentary basins, representing them with simple Gaussians or combinations of Gaussian signals. The same spectral slopes as for gravity signals generated by 3-D fractal source distributions are found for them. Hence the power law decay of the field is not a characteristic only of fractal source models.
If a 3-D fractal source distribution is assumed a priori , a way of verifying the goodness of the model is to examine the whitened field at source level. The probability that the whitened field derives from a random white population is estimated for synthetic and real anomalies by applying the usual statistical tests. 相似文献
We also study the effects of important natural sources, such as salt domes and sedimentary basins, representing them with simple Gaussians or combinations of Gaussian signals. The same spectral slopes as for gravity signals generated by 3-D fractal source distributions are found for them. Hence the power law decay of the field is not a characteristic only of fractal source models.
If a 3-D fractal source distribution is assumed a priori , a way of verifying the goodness of the model is to examine the whitened field at source level. The probability that the whitened field derives from a random white population is estimated for synthetic and real anomalies by applying the usual statistical tests. 相似文献
10.
J. Edwards 《Geophysical Journal International》1982,71(1):199-214
Summary. On the basis of existing theories of rotational remanent magnetization, it is predicted that a gyroremanent magnetization (GRM) might be induced in an isotropic specimen whose orientation is changed during the interval between successive alternating field treatments. An expression is derived, relating the relative magnitude of GRM to the angular displacement, θ of a specimen about an axis perpendicular to the alternating field axis. The predicted effects were observed for four of the five specimens investigated, and in each case GRM was of comparable magnitude to the corresponding partial rotational remanent magnetization. Hysteresis loop curves and characteristics are all consistent with a main magnetic component comprising magnetite grains. A possible connection between GRM and another form of GRM reported in the case of an anisotropic specimen is suggested. Demagnetization of isotropic (and anisotropic) samples by means of the multi-axis technique might be adversely affected by production of GRM. 相似文献
11.
Two approximate methods for weighted principal components analysis (WPCA) were devised and testedin numerical experiments using either empirical variances (obtained from replicated data) or assumedvariances (derived from unreplicated data). In the first ('spherical') approximation each data vector wasassigned a weight proportional to the geometrical mean of its variances in all dimensions. Thearithmetical mean of variances was used instead in the other approximation. Both the numericalexperiments with artificial data containing random errors of various kinds (constant, proportional,constant plus proportional, Poisson) and the analysis of two sets of Raman spectra clearly indicated thenecessity of introducing statistical weights. The spherical approximation was found to be slightly betterthan the arithmetical one. The application of statistical weighting was found to improve the performanceof PCA in estimation problems. 相似文献
12.
Summary. The Cordova gabbro of southern Ontario intrudes 1300 Myr old volcanic rocks of the Hastings Lowlands in the Grenville Structural Province. Three distinct vector magnetizations (A, B and C) have been isolated, using a combination of stable endpoints, subtracted vectors from orthogonal vector plots and converging remagnetization circles. The A magnetization, with mean direction D = 294° I =– 55.5° ( k = 42, α95 = 5.5°, N = 18 sites), is a high coercivity, high blocking temperature remanence recorded by 49 samples. The B magnetization was isolated in 33 samples and has a mean direction D = 305.5° I =– 1.5° ( k = 24, α95 , N = 11 sites). B has lower coercivities and blocking temperatures than A where the two are superimposed. The A and B palaeopoles, 151°E, 10.5°S ( dp = 6°, dm = 8°) and 165.5°E, 24°N ( dp = 5°, dm = 9.5°), fall on the Grenville Track around 900 and 820 Ma respectively. The A and B magnetizations thus date from uplift and cooling following the Grenvillian orogeny. The third magnetization, the C component, has been isolated in 23 samples. Its mean direction is D = 180° I = 27.5° ( k = 18, α95 = 10.5°, N = 12 sites). The C is a low coercivity, low blocking temperature overprint of A and B. Its palaeopole, 102°E, 31°N ( dp = 6.5°, dm = 12°), is unlike post-1300 Precambrian poles for cratonic North America but matches Silurian and late Ordovician poles. 40 Ar/39 Ar plateau ages of 446 and 447 Ma determined by Lopez-Martinez and York for plagioclases from one of the Cordova samples confirm this age assignment. The C magnetization therefore records a previously unrecognized mild thermal or hydrothermal event that occurred in Palaeozoic time, long after the Grenvillian orogeny. 相似文献
13.
Artificial Neural Networks for Mineral-Potential Mapping: A Case Study from Aravalli Province,Western India 总被引：1，自引：0，他引：1
This paper describes a GIS-based application of a radial basis functional link net (RBFLN) to map the potential of SEDEX-type base metal deposits in a study area in the Aravalli metallogenic province (western India). Available public domain geodata of the study area were processed to generate evidential maps, which subsequently were encoded and combined to derive a set of input feature vectors. A subset of feature vectors with known targets (i.e., either known mineralized or known barren locations) was extracted and divided into (a) a training data set and (b) a validation data set. A series of RBFLNs were trained to determine the network architecture and estimate parameters that mapped the maximum number of validation vectors correctly to their respective targets. The trained RBFLN that gave the best performance for the validation data set was used for processing all feature vectors. The output for each feature vector is a predictive value between 1 and 0, indicating the extent to which a feature vector belongs to either the mineralized or the barren class. These values were mapped to generate a predictive classification map, which was reclassified into a favorability map showing zones with high, moderate and low favorability for SEDEX-type base metal deposits in the study area. The method demarcates successfully high favorability zones, which occupy 6% of the study area and contain 94% of the known base metal deposits. 相似文献
14.
A physical model for palaeosecular variation 总被引：1，自引：0，他引：1
Summary. A new model to describe the latitude dependence of the angular dispersion of the palaeomagnetic field (palaeosecular variation) is developed following previous models, but with crucial differences. It is shown that if the probability distribution of virtual geomagnetic poles (VGPs) is circularly symmétric about the rotation axis then the geométry of the distribution of field directions is latitude dependent. This has a significant effect on the latitude dependence of dispersion and is accounted for in the model. The dipole and non-dipole parts of the field are not artificially separated but are intimately linked through an observationally determined relation that the time averaged intensity of the non-dipole field is dependent upon the intensity of the dipole field. It is shown that a consequence of this relation is that no knowledge of the probability distribution of the geomagnetic dipole moment is required. This is a fundamental improvement over previous models.
The model provides excellent fits to the palaeodata and, unlike previous models, is not inconsistent with the latitude variation of the non-dipole field dispersion determined from the present field. For the past 5 Ma the point estimate of the VGP dispersion due to dipole wobble is 7.2° and of the VGP dispersion at the equator due to variation in the non-dipole field is 10.6°. This estimate of the dispersion due to variation in the non-dipole field is in excellent agreement with the value predicted from an analysis of the variation in field intensities over the same period. Fits of the model to data from earlier periods indicate that dispersion due to variation in the non-dipole field is essentially independent of the geomagnetic reversal rate while dipole wobble is positively correlated with reversal rate. 相似文献
The model provides excellent fits to the palaeodata and, unlike previous models, is not inconsistent with the latitude variation of the non-dipole field dispersion determined from the present field. For the past 5 Ma the point estimate of the VGP dispersion due to dipole wobble is 7.2° and of the VGP dispersion at the equator due to variation in the non-dipole field is 10.6°. This estimate of the dispersion due to variation in the non-dipole field is in excellent agreement with the value predicted from an analysis of the variation in field intensities over the same period. Fits of the model to data from earlier periods indicate that dispersion due to variation in the non-dipole field is essentially independent of the geomagnetic reversal rate while dipole wobble is positively correlated with reversal rate. 相似文献
15.
YOSHIKATSU MIYASHITA TOSHIAKI ITOZAWA HIROYUKI KATSUMI SHIN-ICHI SASAKI Department of Knowledge-Based Information Engineering Toyohashi University of Technology Tempaku.Toyohashi Japan 《地理学报(英文版)》1990,(1)
The Non-linear lterative Partial Least Squares(NIPALS)algorithm is used in principal componentanalysis to decompose a data matrix into score vectors and eigenvectors(loading vectors)plus a residualmatrix.N1PALS starts with some guessed starting vector.The principal components obtained by NIPALSdepends on the starting vector;the first principal component could not always be computed.Wold hassuggested a starting vector for NIPALS,but we have found that even if this starting vector is used,thefirst principal component cannot be obtained in all cases.The reason why such a situation occurs isexplained by the power method.A simple modification of the original NIPALS procedure to avoid gettingsmaller eigenvalues is presented. 相似文献
16.
Andrew A. Skabar 《Natural Resources Research》2011,20(3):143-155
Assuming a study region in which each cell has associated with it an N-dimensional vector of values corresponding to N predictor variables, one means of predicting the potential of some cell to host mineralization is to estimate, on the basis
of historical data, a probability density function that describes the distribution of vectors for cells known to contain deposits.
This density estimate can then be employed to predict the mineralization likelihood of other cells in the study region. However,
owing to the curse of dimensionality, estimating densities in high-dimensional input spaces is exceedingly difficult, and
conventional statistical approaches often break down. This article describes an alternative approach to estimating densities.
Inspired by recent work in the area of similarity-based learning, in which input takes the form of a matrix of pairwise similarities
between training points, we show how the density of a set of mineralized training examples can be estimated from a graphical
representation of those examples using the notion of eigenvector graph centrality. We also show how the likelihood for a test
example can be estimated from these data without having to construct a new graph. Application of the technique to the prediction
of gold deposits based on 16 predictor variables shows that its predictive performance far exceeds that of conventional density
estimation methods, and is slightly better than the performance of a discriminative approach based on multilayer perceptron
neural networks. 相似文献
17.
Summary. Procedures are developed for specifying the polarization characteristics of n -dimensional waves, and in particular three-dimensional waves of geophysical interest. We show that when a wave is in a pure state or is totally polarized, all the polarization information can be represented by a single vector u in an n -dimensional unitary space. Simple measures of the degree of polarization of the wave are constructed from the characteristic equation of the spectral matrix S . These measures are functions only of the scalar invariants of S and consequently S need not be diagonalized. If S represents a purely polarized wave, the unitary vector which contains the polarization information about the wave can be obtained directly from S using any 2 n – 1 equations of n 2 possible equations. By multiplying by a phase-factor this unitary vector can be written in the form u = r 1 + i r 2 where r 1 and r 2 are orthogonal vectors in a real space. For an elliptically polarized wave, r 1 and r 2 locate the major and minor axes of the ellipse, and the ellipticity is given by the ratio of their magnitudes. The polarization parameters of ULF magnetic waves at the Earth's surface are computed from one set of five equations ( n = 3) and compared with parameters calculated using established techniques. 相似文献
18.
风力作用下沙粒蠕移概率的转化特征 总被引：3，自引：1，他引：2
建立了沙粒从蠕移状态向静止、跳跃转化以及继续滚动的概率模型.根据滚动沙粒简化的运动方程,结合两个关键随机变量-沙床表面风速和床面位置参数,求出了静止概率、滚动概率、跳跃概率随时间变化的表达式.在此基础上通过对时间求极限得到稳定情况下沙粒处于三种运动状态的概率.利用沙粒蠕移概率除蠕移概率和跃移概率之和,得到沙粒相对蠕移概率,并以此反映蠕移输沙量占总输沙量的比例.计算结果表明,三种转移概率是时间、沙床表面风速和沙粒直径的函数.静止概率随时间增大而增大,但随粒径增大而减小;滚动概率随时间的变化根据粒径的不同表现出不同的特点.粒径较小,滚动概率随时间的增大先增大后减小,存在极大值;粒径较大时,滚动概率随时间的增大逐渐增大,不存在峰值.跳跃概率随时间的增大而增大.三种概率达到稳定状态所需要的时间随粒径的增大而减少.同粒径的沙粒在相同的时间内,三种转移概率会趋于定值.稳定概率和相对蠕移概率由沙床表面风速分布和粒径大小来确定,具有很大的取值范围;静止稳定概率和跳跃稳定概率随粒径的增大分别增大和减小;而滚动稳定概率随粒径的增大逐步先增大后减小.在一定风速条件下,相对蠕移概率随风速和方差的变化都不大,沙粒粒径是最主要的影响因素. 相似文献
19.
Most seismic hazard estimations are based on the assumption of a Poisson process for earthquake occurrence, even though both observations and models indicate a departure of real seismic sequences from this simplistic assumption. Instrumental earthquake catalogues show earthquake clustering on regional scales while the elastic rebound theory predicts a periodic recurrence of characteristic earthquakes on longer timescales for individual events. Recent implementations of time-dependent hazard calculations in California and Japan are based on quasi-periodic recurrences of fault ruptures according to renewal models such as the Brownian Passage Time model. However, these renewal models neglect earthquake interactions and the dependence on the stressing history which might destroy any regularity of earthquake recurrences in reality. To explore this, we investigate the (coupled) stress release model, a stochastic version of the elastic rebound hypothesis. In particular, we are interested in the time-variability of the occurrence of large earthquakes and its sensitivity to the occurrence of Gutenberg–Richter type earthquake activity and fault interactions. Our results show that in general large earthquakes occur quasi-periodically in the model: the occurrence probability of large earthquakes is strongly decreased shortly after a strong event and becomes constant on longer timescales. Although possible stress-interaction between adjacent fault zones does not affect the recurrence time distributions in each zone significantly, it leads to a temporal clustering of events on larger regional scales. The non-random characteristics, especially the quasi-periodic behaviour of large earthquakes, are even more pronounced if stress changes due to small earthquakes are less important. The recurrence-time distribution for the largest events is characterized by a coefficient of variation from 0.6 to 0.84 depending on the relative importance of small earthquakes. 相似文献