Orthogonality of spherical harmonic coefficients     

Malcolm G. McLeod 《Physics of the Earth and Planetary Interiors》1980,23(2):P1-P4

An essentially arbitrary function V(θ, λ) defined on the surface of a sphere can be expressed in terms of spherical harmonics $\text{V(θ, Λ) = a}\text{∑}\text{n=1}\text{∞}\text{∑}\text{m=0}\text{n}\text{p}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{(cos θ) (g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{cos mΛ + h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{sin mΛ)}$ where the P_n^m are the seminormalized associated Legendre polynomials used in geomagnetism, normalized so that $\text{〈[P}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{(cos θ) cos mΛ]〉}{}^2\text{=1/}\text{(2n+1)}$ The angular brackets denote an average over the sphere. The class of functions V(θ, λ) under consideration is that normally of interest in physics and engineering. If we consider an ensemble of all possible orientations of our coordinate system relative to the sphere, then the coefficients g_n^m and h_n^m will be functions of the particular coordinate system orientation, but $\text{〈:(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{〉) = 〈(h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{=}\text{S}{}_{\mathrm{n}}\text{/}\text{(2n=1)}$ where $\text{S}{}_{\mathrm{n}}\text{=}\text{∑}\text{m=0}\text{n}\text{[(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{+ (h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{]}$ for any orientation of the coordinate system (S_n is invariant under rotation of the coordinate system). The averages are over all orientations of the system relative to the sphere. It is also shown that 〈g^m_ng^l_p〉 = 〈h^m_nh^l_p〉 = 0 for l ≠ m or p ≠ n and 〈g^m_nh^l_p〉 = 0 fro all n, m, p, l.  相似文献   

Spatial power spectra of the crustal geomagnetic field and core geomagnetic field     

Malcolm G. McLeod  Paul J. Coleman 《Physics of the Earth and Planetary Interiors》1980,23(2):P5-P19

Lowes (1966, 1974) has introduced the function R_n defined by $\text{R}{}^{\mathrm{n}}\text{=(n + 1)}\text{∑}\text{m=0}\text{∞}\text{[(g}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{+ (h}{}^{\mathrm{m}}{}_{\mathrm{n}}\text{)}{}^2\text{]}\text{where}\text{g}{}_{\mathrm{n}}{}^{\mathrm{m}}\text{and}\text{h}{}_{\mathrm{n}}{}^{\mathrm{m}}$ are the coefficients of a spherical harmonic expansion of the scalar potential of the geomagnetic field at the Earth's surface. The mean squared value of the magnetic field B = ??V on a sphere of radius r > α is given by $\text{〈}\text{B}\text{·〉 =}\text{∑}\text{n=1}\text{∞}\text{R}{}^{\mathrm{n}}\text{(}\text{a/}\text{r}\text{)}{}^{\mathrm{2n=4}}\text{where}\text{a}$ is the Earth's radius. We refer to R_n as the spherical harmonic spatial power spectrum of the geomagnetic field.In this paper it is shown that Rⁿ = R^M_n = R^C_n where the components R_n^M due to the main (or core) field and R_n^C due to the crustal field are given approximately by $\text{R}{}^{\mathrm{M}}{}_{\mathrm{n}}\text{= [}\text{(n =1)/}\text{(n + 2)}\text{](1.142 × 10}{}^9\text{)(0.288}{}^{\mathrm{n}}\text{Λ}{}^2\text{R}{}^{\mathrm{C}}{}_{\mathrm{n}}\text{= [}\text{(n =1){[1 — exp(-n/290)]/}\text{(n/290)} 0.52 Λ}{}^2\text{where}\text{Iγ = 1 nT}$. The two components are approximately equal for n = 15.Lowes has given equations for the core and crustal field spectra. His equation for the crustal field spectrum is significantly different from the one given here. The equation given in this paper is in better agreement with data obtained on the POGO spacecraft and with data for the crustal field given by Alldredge et al. (1963).The equations for the main and crustal geomagnetic field spectra are consistent with data for the core field given by Peddie and Fabiano (1976) and data for the crustal field given by Alldredge et al. The equations are based on a statistical model that makes use of the principle of equipartition of energy and predicts the shape of both the crustal and core spectra. The model also predicts the core radius accurately. The numerical values given by the equations are not strongly dependent on the model.Equations relating average great circle power spectra of the geomagnetic field components to R_n are derived. The three field components are in the radial direction, along the great circle track, and perpendicular to the first two. These equations can, in principle, be inverted to compute the R_n for celestial bodies from average great circle power spectra of the magnetic field components.  相似文献   

Geomagnetic field mapping from a satellite: Spatial power spectra of the geomagnetic field at various satellite altitudes relative to natural noise sources and instrument noise     

Malcolm G. McLeod  Paul J. Coleman 《Physics of the Earth and Planetary Interiors》1980,23(3):222-231

For a low-level geomagnetic satellite survey, for which the motion of the satellite converts spatial variation into temporal variation, the limit on accuracy may well be background temporal fluctuations. The sources of the temporal fluctuations are current systems external to the Earth and include currents induced in the Earth due to these sources. The internal sources consist primarily of two components, the main geomagnetic field with sources in the Earth's core and a crustal geomagnetic field.Power spectra of the vertical geomagnetic field internal component that would be observed by a spacecraft in circular orbit at various altitudes, due to satellite motion through the spatially varying geomagnetic field, are compared to power spectra of the natural temporal fluctuations of the geomagnetic field vertical component (natural noise) and to the power spectrum for typical fluxgate magnetometer instrument noise. The natural noise is shown to be greater than this typical instrument noise over the entire frequency range for which useful measurements of the geomagnetic field may be made, for all geomagnetic latitudes and all times. Thus there would be little benefit in reducing the instrument noise below the typical value of 10^?4 gamma² Hz^?1 plus a 1/f component of 10 milligamma rms decade^?1.For a given satellite altitude, there is a maximum frequency above which the natural noise is greater than the power spectrum of the crustal geomagnetic field vertical component. Below this maximum frequency, the situation is reversed. This maximum frequency depends on geomagnetic latitude (and to a lesser extent on time of day and season of year), being lower in the auroral zone than at lower latitudes. The maximum frequency is also lower at higher satellite altitudes. The maximum frequency determines the spatial resolution obtainable on a magnetic field map. The spatial resolution (for impulses) obtainable at low latitudes for a 100-km satellite altitude (possibly achievable by tethering a small satellite at this altitude to a space vehicle at a higher altitude) is 60 km, while at the auroral zone the obtainable spatial resolution is 100 km. At the higher satellite altitude of 300 km the obtainable spatial resolution is 230 km at low latitudes and 530 km at the auroral zone. At 500-km satellite altitude, the obtainable spatial resolution is 500 km at low latitudes, while maps cannot be made at all for the auroral zone unless the data are selected for “quiet” days.For the lower satellite altitudes, greater spatial resolution can be obtained than at higher altitudes. Furthermore since the crustal geomagnetic field power spectrum is larger at lower altitudes, the relative error due to the natural noise is less than for higher altitudes.  相似文献   

Preliminary observations of kaolinite in a volcanogenic massive sulphide deposit of Permian age     

Dr. R. L. McLeod 《Mineralogy and Petrology》1985,34(3-4):261-269

Summary The Permian Mount Chalmers copper-gold deposit in Central Queensland, Australia, has a mineralogy, form, and setting similar to the kuroko deposits of Japan. Clay observed in the Mount Chalmers deposit is considered to have formed by subsea-floor replacement of the footwall tuffs and dolomitic horizons, and by direct precipitation from the ore solutions on the sea-floor near the fumarolic orifice.X-ray powder diffraction study of the clay indicates the mineral is kaolinite, rather than another member of the kandite group. Resolution of the and doublet in the diffraction trace of this clay shows that the mineral is well-crystallized. Electron microprobe analysis indicates that some samples of this clay approach the theoretical chemical composition, while others carry minor amounts of impurity elements such as Fe, Mg, Ca, K, and Na. SiO₂:Al₂O₃ mole ratio for this kaolinite is similar to values reported in the literature.

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Kaolinite dans un gisement de sulphides Permien: Observations préliminaires

Résumé Le gisement de cuivre et d'or du Mont Chalmers Permien au centre du Queensland en Australie, a une minéralogie, une forme et une disposition similaire au gisement kuroko au Japon. On considère que les argiles observées au gisement Mont Chalmers ont été formées par le remplacement du fond sousmarine du sol tufeux et des couches dolomitiques et par la précipitation directe des solutions minérales sur le fond de la mer au voisinage de la fumerolle.L'étude par diffraction radiographique de la poudre d'argile montre que le minerai est du kaolinite, plutôt qu'un autre membre du groupe kaolinite. La résolution du doublet et dans le tracé de la diffraction de cette argile montre que le minerai est bien cristallisé. L'analyse par microsonde électronique indique que quelques échantillons de cette argile s'approchent de la composition chimique théorique, tandis que les autres contiennent des traces d'impuretés d'éléments tels que Fe, Mg, Ca, K et Na. La proportion moléculaire SiO₂:Al₂ O₃ de ce kaolinite est similaire aux valeurs présentées dans la littérature.

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With 3 Figures  相似文献   

The Castles Project     

J. A. Muñoz  E. E. Falco  C. S. Kochanek  J. Lehár  B. A. McLeod  C. D. Impey  H.-W. Rix  C. Y. Peng 《Astrophysics and Space Science》1998,263(1-4):51-54

We describe the goals of the CASTLES (CfA-Arizona-Space-Telescope-LEns-Survey) project including a sample of NICMOS images of gravitational lenses and a brief list of the preliminary findings. This revised version was published online in July 2006 with corrections to the Cover Date.  相似文献   

New metrics for managing and sustaining the ocean's bounty     

Heather Tallis  Sarah E. LesterMary Ruckelshaus  Mark PlummerKaren McLeod  Anne GuerrySandy Andelman  Margaret R. CaldwellMarc Conte  Stephen CoppsDavid Fox  Rod FujitaSteven D. Gaines  Guy GelfenbaumBarry Gold  Peter KareivaChoong-ki Kim  Kai LeeMichael Papenfus  Scott RedmanBrian Silliman  Lisa WaingerCrow White 《Marine Policy》2012,36(1):303-306

Policies are arising around the world, most recently in the United States, that mandate the implementation of marine spatial planning as a practical pathway towards ecosystem-based management. In the new United States ocean policy, and several other cases around the globe, ecosystem services are at the core of marine spatial planning, but there is little guidance on how ecosystem services should be measured, making it hard to implement this new approach. A new framework is shown here for practical, rigorous ecosystem service measurement that highlights contributions from both natural and social systems. The novel three-step framework addresses traditional shortcomings of an ecosystem services approach by giving managers and scientists the tools to assess and track: (1) the condition of the ecosystem (supply metrics), (2) the amount of ocean resources actually used or enjoyed by people (service metrics), and (3) people's preference for that level of service (value metrics). This framework will allow real world progress on marine spatial planning to happen quickly, and with a greater chance for success.  相似文献   

Kalman filter estimation for periodic autoregressive-moving average models     

C. Jimenez  A. I. McLeod  K. W. Hipel 《Stochastic Environmental Research and Risk Assessment (SERRA)》1989,3(3):227-240

An exact maximum likelihood procedure is presented for estimating the parameters of a periodic autogressive-moving average (PARMA) model. To develop an estimator which is both statistically and computationally efficient, the PARMA class of models is written using a state-space representation and a Kalman filtering algorithm is used to estimate the parameters. In order to demonstrate how to fit PARMA models in practice, the most appropriate types of PARMA models are identified for fitting to two average monthly riverflow time series and the new estimator is employed for estimating the model parameters.  相似文献   

Extended search for correlation between solar mean magnetic field Bison data and coronal mass ejections     

William J. Chaplin  Andrew M. Dumbill  Yvonne P. Elsworth  George R. Isaak  Clive P. McLeod  Brek A. Miller  Roger New  Balázs Pintér 《Solar physics》2004,220(2):307-316

The Birmingham Solar-Oscillations Network (BiSON) has acquired high-precision solar mean magnetic field (SMMF) data on a 40-s cadence for a decade. We present attempts to compare such data from recent years with the occurrence of coronal mass ejections (CMEs) as recorded by LASCO, using correlation techniques applied to measurements from different BiSON instruments to maximise the sensitivity to CME-related SMMF responses. SMMF measurements were recorded at the time of occurrence of several hundred CMEs. No CME event shows a convincing response in our SMMF data at short periods setting a threshold amplitude of 12 mG. By averaging data sets we are able to set lower thresholds, which depend somewhat on the distribution of response strengths. A brief summary of the very first results of this study is also given in Chaplin et al.  相似文献   

锂辉石:巨型花岗伟晶岩锂-铯-钽矿床中关键含锂矿物SCIEI北大核心CSCD     

DI GROVES  Liang Zhang  IM GROVES  AK SENER 《岩石学报》2022,38(1):1-8

寻找传统化石燃料的替代能源已成为全球性议题。受动力电池消费的拉动,锂资源需求急剧上升,伟晶岩型锂矿勘查热度持续攀升。虽然众多伟晶岩型锂矿地质特征尚不清晰,已有证据表明锂辉石是大多数大型-巨型伟晶岩型锂矿床的主要含锂矿物。与许多近直立的伟晶岩脉群不同,世界范围内大多数太古代伟晶岩矿脉往往呈近水平或缓倾斜在角闪岩相围岩中产出,它们往往具有复杂的三维形态并发育明显的矿物和地球化学分带。这些太古代伟晶岩脉通常形成于挤压或压剪构造体制下同变质环境中,成岩期最小主应力(σ;)近竖直。因此,伟晶岩常常侵位于近水平的构造局部引张区而形成复杂的几何学形态。压性的构造环境为富锂熔体多次脉动式注入和富含挥发分熔体垂向结晶分异提供了充足的时间;锂辉石在中高温压条件下结晶成为缓倾富锂带中最为常见的含锂矿物。  相似文献   

Multivariate contemporaneous ARMA model with hydrological applications     

F. Camacho  A. I. McLeod  K. W. Hipel 《Stochastic Environmental Research and Risk Assessment (SERRA)》1987,1(2):141-154

In order to allow contemporaneous autoregressive moving average (CARMA) models to be properly applied to hydrological time series, important statistical properties of the CARMA family of models are developed. For calibrating the model parameters, efficient joint estimation procedures are investigated and compared to a set of uivariate estimation procedures. It is shown that joint estimation procedures improve the efficiency of the autoregressive and moving average parameter estimates, but no improvements are expected on the estimation of the mean vector and the variance covariance matrix of the model. The effects of the different estimation procedures on the asymptotic prediction error are also considered. Finally, hydrological applications demonstrate the usefulness of the CARMA models in the field of water resources.  相似文献