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At least 145 records · Page 8

Is the Non-Dipole Magnetic Field Random?

Statistical modelling of the Earth's magnetic field B has a long history. In particular, the spherical harmonic coefficients of scalar fields derived from B can be treated as Gaussian random variables. In this paper, we give examples of highly organized fields whose spherical harmonic coefficients pass tests for independent Gaussian random variables. The fact that coefficients at some depth may be usefully summarized as independent samples from a normal distribution need not imply that there really is some physical, random process at that depth. In fact, the field can be extremely structured and still be regarded for some purposes as random. In this paper, we examined the radial magnetic field B(sub r) produced by the core, but the results apply to any scalar field on the core-mantle boundary (CMB) which determines B outside the CMB.

Walker, Andrew D.↗

The GRACE Mission: Meeting the Technical Challenges

The Gravity Recovery and Climate Experiment (GRACE) Mission is scheduled for launch in June 2001. Within the first year of the GRACE Mission, the project has a minimum science requirement to deliver a new model of the Earth's static geoid with an error of less than 1 cm to spherical harmonic degree seventy (70). However, the performance of the GRACE Mission is designed to exceed this minimum requirement by a factor of 25 or more. For spherical harmonic degrees of up to 40, we expect to improve the current knowledge of the gravity field by one thousand (1000x). The GRACE Mission uses the satellite-to-satellite tracking (SST) technique. The twin GRACE satellites are the instruments that measure the nonuniformities in the Earth's gravity field. Nonuniformities in the gravity field cause the relative distance between the centers-of-mass of the two satellites to vary as they fly over the Earth. Atmospheric drag is the largest non-gravitational disturbing force. Drag is measured and will be used to correct changes in the satellite-to-satellite range measured by an SST microwave link. The microwave link will measure changes in the range between the two GRACE satellites with an error approaching 1 micron. We will discuss how these instrumentation requirements affect the configuration, the mass balance, the thermal control and the aerodynamic design of the satellites, and the design of the microwave SST link and the accelerometer. Finally, the question of how noise in these components limits the overall accuracy of the gravity models will be addressed.

Davis, E. S.↗

Feeling Gravity's Pull: Gravity Modeling. The Gravity Field of Mars

Most people take the constant presence of gravitys pull for granted. However, the Earth's gravitational strength actually varies from location to location. This variation occurs because mass, which influences an object's gravitational pull, is not evenly distributed within the planet. Changes in topography, such as glacial movement, an earthquake, or a rise in the ocean level, can subtly affect the gravity field. An accurate measurement of the Earth's gravity field helps us understand the distribution of mass beneath the surface. This insight can assist us in locating petroleum, mineral deposits, ground water, and other valuable substances. Gravity mapping can also help notice or verify changes in sea surface height and other ocean characteristics. Such changes may indicate climate change from polar ice melting and other phenomena. In addition, gravity mapping can indicate how land moves under the surface after earthquakes and other plate tectonic processes. Finally, changes in the Earth's gravity field might indicate a shift in water distribution that could affect agriculture, water supplies for population centers, and long-term weather prediction. Scientists can map out the Earth's gravity field by watching satellite orbits. When a satellite shifts in vertical position, it might be passing over an area where gravity changes in strength. Gravity is only one factor that may shape a satellite's orbital path. To derive a gravity measurement from satellite movement, scientists must remove other factors that might affect a satellite's position: 1. Drag from atmospheric friction. 2. Pressure from solar radiation as it heads toward Earth and. as it is reflected off the surface of the Earth 3. Gravitational pull from the Sun, the Moon, and other planets in the Solar System. 4. The effect of tides. 5. Relativistic effects. Scientists must also correct for the satellite tracking process. For example, the tracking signal must be corrected for refraction through the atmosphere of the Earth. Supercomputers can calculate the effect of gravity for specific locations in space following a mathematical process known as spherical harmonics, which quantifies the gravity field of a planetary body. The process is based on Laplace's fundamental differential equation of gravity. The accuracy of a spherical harmonic solution is rated by its degree and order. Minute variations in gravity are measured against the geoid, a surface of constant gravity acceleration at mean sea level. The geoid reference gravity model strength includes the central body gravitational attraction (9.8 m/sq s) and a geopotential variation in latitude partially caused by the rotation of the Earth. The rotational effect modifies the shape of the geoid to be more like an ellipsoid, rather than a perfect, circle. Variations of gravity strength from the ellipsoidal reference model are measured in units called milli-Galileos (mGals). One mGal equals 10(exp -5) m/sq s. Research projects have also measured the gravity fields of other planetary bodies, as noted in the user profile that follows. From this information, we may make inferences about our own planet's internal structure and evolution. Moreover, mapping the gravity fields of other planets can help scientists plot the most fuel-efficient course for spacecraft expeditions to those planets.

Lemoine, Frank↗

Lateral density anomalies and the earth's gravitational field

The interpretation of gravity is valuable for understanding lithospheric plate motion and mantle convection. Postulated models of anomalous mass distributions in the earth and the observed geopotential as expressed in the spherical harmonic expansion are compared. In particular, models of the anomalous density as a function of radius are found which can closely match the average magnitude of the spherical harmonic coefficients of a degree. These models include: (1) a two-component model consisting of an anomalous layer at 200 km depth (below the earth's surface) and at 1500 km depth (2) a two-component model where the upper component is distributed in the region between 1000 and 2800 km depth, and(3) a model with density anomalies which continuously increase with depth more than an order of magnitude.

Lowrey, B. E.↗

High-resolution Local Gravity Model of the South Pole of the Moon from GRAIL Extended Mission Data

We estimated a high-resolution local gravity field model over the south pole of the Moon using data from the Gravity Recovery and Interior Laboratory's extended mission. Our solution consists of adjustments with respect to a global model expressed in spherical harmonics. The adjustments are expressed as gridded gravity anomalies with a resolution of 1/6deg by 1/6deg (equivalent to that of a degree and order 1080 model in spherical harmonics), covering a cap over the south pole with a radius of 40deg. The gravity anomalies have been estimated from a short-arc analysis using only Ka-band range-rate (KBRR) data over the area of interest. We apply a neighbor-smoothing constraint to our solution. Our local model removes striping present in the global model; it reduces the misfit to the KBRR data and improves correlations with topography to higher degrees than current global models.

South Pole↗

Spatial and temporal variations of monthly mean total columnar ozone derived from 7 years of BUV data

Ninth-degree spherical harmonic models are used to represent monthly mean total columnar ozone fields, with spatial spectra defined in terms of simple analytic expressions representing the rms value of each spherical harmonic coefficient. The spectra are combined with estimation theory techniques to develop a realistic error analysis accounting for errors due to nonglobal coverage. Amplitude and phase variations for longitudinal waves one, two and three are presented, and it is shown that these waves reinforce in space and time in the period of high poleward ozone transport in the northern hemisphere. In the southern hemisphere, the wave two and three components are much smaller, with wave one maximum amplitude occurring in September and October.

Tolson, R. H.↗

Precise solution for a finite set of spherical coefficients from equiangular gridded data

An important goal of geodesy is to determine the anomalous potential and its derivatives outside of the earth. Representing the surface anomalies by a series of spherical harmonics is useful since it is then possible to do a term by term solution of Laplace's equation and upward continuation. The problem of finding such a spherical harmonic series for anomaly values given on an equiangular surface grid is addressed. (This is a first step toward the more complicated problem of finding a function such that locally averaged values fit a grid of mean anomalies.) Three approaches to this fitting problem are discussed and compared: the discrete Fourier technique, the discrete integral technique, and a new approach. The peculiar nature of the equiangular grid, with its increasing density of (noisy) data toward the poles, causes each method to exhibit a different type of difficulty. The new method is shown to be practical as well as precise since the numerical conditioning problems which appear can be successfully handled by such well-known techniques as a (simple) Kalman filter.

Zucker, Paul A.↗

Decoding Mode-mixing in Black-hole Merger Ringdown

Optimal extraction of information from gravitational-wave observations of binary black-hole coalescences requires detailed knowledge of the waveforms. Current approaches for representing waveform information are based on spin-weighted spherical harmonic decomposition. Higher-order harmonic modes carrying a few percent of the total power output near merger can supply information critical to determining intrinsic and extrinsic parameters of the binary. One obstacle to constructing a full multi-mode template of merger waveforms is the apparently complicated behavior of some of these modes; instead of settling down to a simple quasinormal frequency with decaying amplitude, some |m| = modes show periodic bumps characteristic of mode-mixing. We analyze the strongest of these modes the anomalous (3, 2) harmonic mode measured in a set of binary black-hole merger waveform simulations, and show that to leading order, they are due to a mismatch between the spherical harmonic basis used for extraction in 3D numerical relativity simulations, and the spheroidal harmonics adapted to the perturbation theory of Kerr black holes. Other causes of mode-mixing arising from gauge ambiguities and physical properties of the quasinormal ringdown modes are also considered and found to be small for the waveforms studied here.

Kelly, Bernard J.↗

An extended Vlasov–Fokker–Planck approach for kinetic simulations of laser plasmas

Vlasov–Fokker–Planck simulation codes occupy an important niche in modeling laser-produced plasmas, since they are well suited to studying the effect of collisions on electron kinetic phenomena, especially energy transport. One of the most important elements of energy transport is the absorption of laser light by the plasma; however, simulating this in detail requires resolving oscillations of the laser light, whose characteristic timescale is orders of magnitude shorter than the simulation time needed to study transport physics. For this reason, most Vlasov–Fokker–Planck codes used to study electron transport in laser plasmas rely on simplified models of the laser–plasma coupling. Their underlying assumptions nominally preclude their use for modeling laser light having short-scale structure in space or time, such as broadband lasers. In this work, we derive a more general computational framework suitable for arbitrarily structured laser fields. Furthermore, our approach is based on an extended set of Vlasov–Fokker–Planck equations that separately solve for the low- and high-frequency plasma response. We implement these extended Vlasov–Fokker–Planck equations in the spherical harmonic code K2 and demonstrate the performance of the method on several laser absorption test problems, with particular attention to the judicious selection of time steps, time integrators, and spherical harmonic truncation, according to the intensity and spectrum of the laser light under consideration. Comparison with the widely used Langdon absorption operator shows the Langdon operator performs remarkably well for predicting laser heating in the simple cases considered here, even in situations that would seem to violate its underlying assumptions.

70 PLASMA PHYSICS AND FUSION TECHNOLOGY↗

High resolution mapping of the magnetic field of the solar corona

The mapping of the current-free magnetic field of the solar corona and the reliability of the spherical harmonic analysis of the photospheric magnetic field pattern are improved by data with much greater dynamic range and spatial resolution than previously available and a new algorithm which permits spherical harmonic expansion to a much higher value of the principal index. Coronal field maps can be drawn for local regions, for just the open field lines, and for various spatial resolutions on a global scale.

Altschuler, M. D.↗

On The Jacobian of the ECEF J2 Gravitation Model

An Earth-centered, Earth-fixed (ECEF) inertial navigation system must compute the Jacobian of its employed gravitation model with respect to position while time-propagating the error covariance of the system. One commonly used gravitation model is the ‘J2 model’ which is a second-order truncation of the Earth’s spherical harmonic gravitation model. The J2 model is popular because it can quickly and efficiently be evaluated, and the truncation error is small: The ‘J3 term’ --- the third term in the spherical harmonic expansion --- is approximately 1000 times smaller than the J2 term.

42 ENGINEERING↗

Venus - Global gravity and topography

A new gravity field determination that has been produced combines both the Pioneer Venus Orbiter (PVO) and the Magellan Doppler radio data. Comparisonsbetween this estimate, a spherical harmonic model of degree and order 21, and previous models show that significant improvements have been made. Results are displayed as gravity contours overlaying a topographic map. We also calculate a new spherical harmonic model of topography based on Magellan altimetry, with PVO altimetry included where gaps exist in the Magellan data. This model is also of degree and order 21, so in conjunction with the gravity model, Bouguer and isostatic anomaly maps can be produced. These results are very consistent with previous results, but reveal more spatial resolution in the higher latitudes.

Mcnamee, J. B.↗

Terrestrial Water Mass Load Changes from Gravity Recovery and Climate Experiment (GRACE)

Recent studies show that data from the Gravity Recovery and Climate Experiment (GRACE) is promising for basin- to global-scale water cycle research. This study provides varied assessments of errors associated with GRACE water storage estimates. Thirteen monthly GRACE gravity solutions from August 2002 to December 2004 are examined, along with synthesized GRACE gravity fields for the same period that incorporate simulated errors. The synthetic GRACE fields are calculated using numerical climate models and GRACE internal error estimates. We consider the influence of measurement noise, spatial leakage error, and atmospheric and ocean dealiasing (AOD) model error as the major contributors to the error budget. Leakage error arises from the limited range of GRACE spherical harmonics not corrupted by noise. AOD model error is due to imperfect correction for atmosphere and ocean mass redistribution applied during GRACE processing. Four methods of forming water storage estimates from GRACE spherical harmonics (four different basin filters) are applied to both GRACE and synthetic data. Two basin filters use Gaussian smoothing, and the other two are dynamic basin filters which use knowledge of geographical locations where water storage variations are expected. Global maps of measurement noise, leakage error, and AOD model errors are estimated for each basin filter. Dynamic basin filters yield the smallest errors and highest signal-to-noise ratio. Within 12 selected basins, GRACE and synthetic data show similar amplitudes of water storage change. Using 53 river basins, covering most of Earth's land surface excluding Antarctica and Greenland, we document how error changes with basin size, latitude, and shape. Leakage error is most affected by basin size and latitude, and AOD model error is most dependent on basin latitude.

Seo, K.-W.↗

Parameter estimation applied to Nimbus 6 wide-angle longwave radiation measurements

A parameter estimation technique was used to analyze the August 1975 Nimbus 6 Earth radiation budget data to demonstrate the concept of deconvolution. The longwave radiation field at the top of the atmosphere is defined from satellite data by a fifth degree and fifth order spherical harmonic representation. The variations of the major features of the radiation field are defined by analyzing the data separately for each two-day duty cycle. A table of coefficient values for each spherical harmonic representation is given along with global mean, gradients, degree variances, and contour plots. In addition, the entire data set is analyzed to define the monthly average radiation field.

Green, R. N.↗

Geomagnetic temporal change: 1903-1982 - A spline representation

The secular variation of the earth's magnetic field is itself subject to temporal variations. These are investigated with the aid of the coefficients of a series of spherical harmonic models of secular variation deduced from data for the interval 1903-1982 from the worldwide network of magnetic observatories. For some studies it is convenient to approximate the time variation of the spherical harmonic coefficients with a smooth, continuous, function; for this a spline fitting is used. The phenomena that are investigated include periodicities, discontinuities, and correlation with the length of day. The numerical data presented will be of use for further investigations and for the synthesis of secular variation at any place and at any time within the interval of the data - they are not appropriate for temporal extrapolations.

Langel, R. A.↗

Investigation of a geodesy coexperiment to the Gravity Probe B relativity gyroscope program

Geodesy is the science of measuring the gravitational field of and positions on the Earth. Estimation of the gravitational field via gravitation gradiometry, the measurement of variations in the direction and magnitude of gravitation with respect to position, is this dissertation's focus. Gravity Probe B (GP-B) is a Stanford satellite experiment in gravitational physics. GP-B will measure the precession the rotating Earth causes on the space time around it by observing the precessions of four gyroscopes in a circular, polar, drag-free orbit at 650 km altitude. The gyroscopes are nearly perfect niobium-coated spheres of quartz, operating at 1.8 K to permit observations with extremely low thermal noise. The permissible gyroscope drift rate is miniscule, so the torques on the gyros must be tiny. A drag-free control system, by canceling accelerations caused by nongravitational forces, minimizes the support forces and hence torques. The GP-B system offers two main possibilities for geodesy. One is as a drag-free satellite to be used in trajectory-based estimates of the Earth's gravity field. We described calculations involving that approach in our previous reports, including comparison of laser only, GPS only, and combined tracking and a preliminary estimate of the possibility of estimating relativistic effects on the orbit. The second possibility is gradiometry. This technique has received a more cursory examination in previous reports, so we concentrate on it here. We explore the feasibility of using the residual suspension forces centering the GP-B gyros as gradiometer signals for geodesy. The objective of this work is a statistical prediction of the formal uncertainty in an estimate of the Earth's gravitation field using data from GP-B. We perform an instrument analysis and apply two mathematical techniques to predict uncertainty. One is an analytical approach using a flat-Earth approximation to predict geopotential information quality as a function of spatial wavelength. The second estimates the covariance matrix arising in a least-squares estimate of a spherical harmonic representation of the geopotential using GP-B gradiometer data. The results show that the GP-B data set can be used to create a consistent estimate of the geopotential up to spherical harmonic degree and order 60. The formal uncertainty of all coefficients between degrees 5 and 50 is reduced by factors of up to 30 over current satellite-only estimates and up to 7 over estimates which include surface data. The primary conclusion resulting from this study is that the gravitation gradiometer geodesy coexperiment to GP-B is both feasible and attractive.

Everitt, C. W. F.↗

Deconvolution of wide field-of-view radiometer measurements of earth-emitted radiation. II - Analysis of first year of Nimbus 6 ERB data

The theory of deconvolution considered by Smith and Green (1981) is applied to Nimbus 6 data in order to interpret the data with the maximum obtainable resolution. Attention is given to the data sample, sun-contaminated measurements, the measurement model, the deconvolution, the degree variance, the spherical harmonic coefficients, the geographical distribution of longwave radiation, time histories of zonal coefficients, and the effect of a grid system. Degree variance plots for 12 months of longwave radiation data show that the limit for a spherical harmonic representation of the Nimbus 6 wide field-of-view longwave data is 12th degree. The degree variance plots also reveal that most of the power is in the lower degree terms. Contour maps of the radiation field show the geographical distribution of earth emitted radiant exitance for each month.

Bess, T. D.↗

Continental and oceanic crustal magnetization modelling

Inversion of magnetic data from the MAGSAT satellite, to arrive at intensities of magnetization of the Earth's crust, was performed by two different methods. The first method uses a spherical harmonic model of the magnetic field. The coefficients believed to represent sources in the Earth's crust can then be inverted to arrive at vertical dipole moments per unit area at the Earth's surface. The spherical harmonic models contain coefficients of degrees of harmonics up to 23. The dipole moment per unit area for a surface element can then be determined by summing the contribution for each individual degree of harmonic. The magnetic moments were calculated for continental and oceanic areas separately as well as over certain latitudinal segments. Of primary concern was to determine whether there are any differences between continental and oceanic areas. The second analysis with magnetization intensities was made using narrower ranges of degrees of harmonics, assuming that higher degrees are present in the core field signal.

Harrison, C. G. A.↗