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All sourcesNASA NTRSDOE OSTI
NASA NTRS1976

Spherical harmonic representation of the gravity field of Mars using a short-arc technique

Planetary gravity fields are primarily determined from Doppler tracking of orbiting spacecraft which are typically in high eccentricity orbits. Thus, the non-central gravitational components perturb the orbit primarily in the vicinity of periapsis. To find a computationally efficient technique and to reduce the influence of unmodeled forces on the spacecraft, an analysis has been performed of the effectiveness of processing only data in the vicinity of periapsis using multiple short arcs. Mariner 9 data are processed for the study, using 1, 2, 3 and 4 hour arcs and estimating gravity fields from 4th through 7th degree. The resulting gravity fields are inter-compared and also compared with previously published results.

Daniels, E. F.↗
NASA NTRS1981

Spherical harmonic representation of the main geomagnetic field for world charting and investigations of some fundamental problems of physics and geophysics

Quiet-day data from MAGSAT were examined for effects which might test the validity of Maxwell's equations. Both external and toroidal fields which might represent a violation of the equations appear to exist, well within the associated errors. The external field might be associated with the ring current, and varies of a time-scale of one day or less. Its orientation is parallel to the geomagnetic dipole. The toriodal field can be confused with an orientation in error (in yaw). It the toroidal field really exists, its can be related to either ionospheric currents, or to toroidal fields in the Earth's core in accordance with Einstein's unified field theory, or to both.

Barraclough, D. R.↗
NASA NTRS1982

Spherical harmonic representation of the main geomagnetic field for world charting and investigations of some fundamental problems of physics and geophysics

Progress in the harmonic analysis of MAGSAT data is reported. Single-day data sets were subdivided into information on the sunrise side of the Earth and information on the sunset side of the Earth. Data for the main and external fields each demonstrate a clear and consistent systematic difference between the sets of data which was determined to be, due to ionospheric currents which differ from the sunset to the sunrise terminator. A toroidal field was analyzed for and determined to be an apparent toroidal field resulting from electric currents concentrated in the two terminators. Progressive elimination of auroral zone data demonstrates that the information presented does not arise from complications due to Birkeland currents.

Barraclough, D. R.↗
NASA NTRS1986

Spherical harmonic expansions of the Earth's gravitational potential to degree 360 using 30' mean anomalies

Two potential coefficient fields that are complete to degree and order 360 have been computed. One field (OSU86E) excludes geophysically predicted anomalies while the other (OSU86F) includes such anomalies. These fields were computed using a set of 30' mean gravity anomalies derived from satellite altimetry in the ocean areas and from land measurements in North America, Europe, Australia, Japan and a few other areas. Where no 30' data existed, 1 deg x 1 deg mean anomaly estimates were used if available. No rigorous combination of satellite and terrestrial data was carried out. Instead advantage was taken of the adjusted anomalies and potential coefficients from a rigorous combination of the GEML2' potential coefficient set and 1 deg x 1 deg mean gravity anomalies. The two new fields were computed using a quadrature procedure with de-smoothing factors. The spectra of the new fields agree well with the spectra of the fields with 1 deg x 1 deg data out to degree 180. Above degree 180 the new fields have more power. The fields have been tested through comparison of Doppler station geoid undulations with undulations from various geopotential models. The agreement between the two types of undulations is approximately + or - 1.6 m. The use of a 360 field over a 180 field does not significantly improve the comparison. Instead it allows the comparison to be done at some stations where high frequency effects are important. In addition maps made in areas of high frequency information (such as trench areas) clearly reveal the signal in the new fields from degree 181 to 360.

Rapp, Richard H.↗
NASA NTRS1988

On estimating the basin-scale ocean circulation from satellite altimetry. Part 1: Straightforward spherical harmonic expansion

Direct estimation of the absolute dynamic topography from satellite altimetry has been confined to the largest scales (basically the basin-scale) owing to the fact that the signal-to-noise ratio is more unfavorable everywhere else. But even for the largest scales, the results are contaminated by the orbit error and geoid uncertainties. Recently a more accurate Earth gravity model (GEM-T1) became available, providing the opportunity to examine the whole question of direct estimation under a more critical limelight. It is found that our knowledge of the Earth's gravity field has indeed improved a great deal. However, it is not yet possible to claim definitively that our knowledge of the ocean circulation has improved through direct estimation. Yet, the improvement in the gravity model has come to the point that it is no longer possible to attribute the discrepancy at the basin scales between altimetric and hydrographic results as mostly due to geoid uncertainties. A substantial part of the difference must be due to other factors; i.e., the orbit error, or the uncertainty of the hydrographically derived dynamic topography.

Tai, Chang-Kou↗
NASA NTRS1999

Artifacts Introduced in the Point Evaluation of Functions Expanded into a Degree 360 Spherical Harmonic Series

An expansion of a function initially given in 1deg cells was carried out to degree 360 by using 30'cells whose value was initially assigned to be the value of the 1deg cell in which it fell. The evaluation of point values of the function from the degree 360 expansion revealed spurious patterns attributed to the coefficients from degree 181 to 360. Expansion of the original function in 1deg cells to degree 180 showed no problems in the point evaluation. Mean 1deg values computed from both degree 180 to 360 expansions showed close agreement with the original function. The artifacts could be removed if the 30' values were interpolated by spline procedures from adjacent I' cells. These results led to an examination of the gravity anomalies and geoid undulations from EGM96 in areas where I' values were "split up" to form 30'cells. The area considered was 75degS to 85degS, 100degE to 120degE where the split up cells were basically south of 81 degS. A small, latitude related, and possibly spurious effect might be detectable in anomaly variations in the region. These results suggest that point values of a function computed from a high degree expansion may have spurious signals unless the cell size is compatible with the maximum degree of expansion. The spurious signals could be eliminated by using a spline interpolation procedure to obtain the 30'values from the 1deg values.

Rapp, R.↗
NASA NTRS1981

Role of membrane stresses in the support of planetary topography

The role of membrane stresses and bending stresses in supporting topographic loads on planetary elastic lithospheres is examined. A dimensionless parameter is introduced in order to determine the ability of a spherical shell to support loads through membrane stresses. It is determined that when this parameter is large, membrane stresses can fully support topographic loads with flexure, and when it is small the influence of the membrane stresses can be neglected. Equations governing the behavior of a spherical shell are solved for a topographic load expressed in terms of spherical harmonics, and spherical harmonic expansions of the measured gravity and topography for Mars and the moon are compared with the theory. It is concluded that membrane stresses play an important role in the support of topographic loads on the moon and Mars. The correlation of observed gravitational potential anomalies with the topography on Mars is explained by membrane stresses in the elastic lithosphere.

Turcotte, D. L.↗
NASA NTRS1994

Spherical disharmonics in the Earth sciences and the spatial solution: Ridges, hotspots, slabs, geochemistry and tomography correlations

There is increasing use of statistical correlations between geophysical fields and between geochemical and geophysical fields in attempts to understand how the Earth works. Typically, such correlations have been based on spherical harmonic expansions. The expression of functions on the sphere as spherical harmonic series has many pitfalls, especially if the data are nonuniformly and/or sparsely sampled. Many of the difficulties involved in the use of spherical harmonic expansion techniques can be avoided through the use of spatial domain correlations, but this introduces other complications, such as the choice of a sampling lattice. Additionally, many geophysical and geochemical fields fail to satisfy the assumptions of standard statistical significance tests. This is especially problematic when the data values to be correlated with a geophysical field were collected at sample locations which themselves correlate with that field. This paper examines many correlations which have been claimed in the past between geochemistry and mantle tomography and between hotspot, ridge, and slab locations and tomography using both spherical harmonic coefficient correlations and spatial domain correlations. No conclusively significant correlations are found between isotopic geochemistry and mantle tomography. The Crough and Jurdy (short) hotspot location list shows statistically significant correlation with lowermost mantle tomography for degree 2 of the spherical harmonic expansion, but there are no statistically significant correlations in the spatial case. The Vogt (long) hotspot location list does not correlate with tomography anywhere in the mantle using either technique. Both hotspot lists show a strong correlation between hotspot locations and geoid highs when spatially correlated, but no correlations are revealed by spherical harmonic techniques. Ridge locations do not show any statistically significant correlations with tomography, slab locations, or the geoid; the strongest correlation is with lowermost mantle tomography, which is probably spurious. The most striking correlations are between mantle tomography and post-Pangean subducted slabs. The integrated locations of slabs correlate strongly with fast areas near the transition zone and the core-mantle boundary and with slow regions from 1022-1248 km depth. This seems to be consistent with the 'avalanching' downwellings which have been indicated by models of the mantle which include an endothermic phase transition at the 670-km discontinuity, although this is not a unique interpretation. Taken as a whole, these results suggest that slabs and associated cold downwellings are the dominant feature of mantle convection. Hotspot locations are no better correlated with lower mantle tomography than are ridge locations.

Ray, Terrill W.↗
NASA NTRS2011

What Supergranule Flow Models Tell Us About the Sun's Surface Shear Layer and Magnetic Flux Transport

Models of the photospheric flows due to supergranulation are generated using an evolving spectrum of vector spherical harmonics up to spherical harmonic wavenumber l~1500. Doppler velocity data generated from these models are compared to direct Doppler observations from SOHO/MDI and SDO/HMI. The models are adjusted to match the observed spatial power spectrum as well as the wavenumber dependence of the cell lifetimes, differential rotation velocities, meridional flow velocities, and relative strength of radial vs. horizontal flows. The equatorial rotation rate as a function of wavelength matches the rotation rate as a function of depth as determined by global helioseismology. This leads to the conclusions that the cellular structures are anchored at depths equal to their widths, that the surface shear layer extends to at least 70 degrees latitude, and that the poleward meridional flow decreases in amplitude and reverses direction at the base of the surface shear layer (approx.35 Mm below the surface). Using the modeled flows to passively transport magnetic flux indicates that the observed differential rotation and meridional flow of the magnetic elements are directly related to the differential rotation and meridional flow of the convective pattern itself. The magnetic elements are transported by the evolving boundaries of the supergranule pattern (where the convective flows converge) and are unaffected by the weaker flows associated with the differential rotation or meridional flow of the photospheric plasma.

Hathaway, David↗
NASA NTRS2010

Evaluating Descent and Ascent Trajectories Near Non-Spherical Bodies

Spacecraft landing on small bodies pass through regions where conventional gravitation formulations using exterior spherical harmonics are inaccurate. An investigation shows that a formulation using interior solid spherical harmonics might be satisfactory. Interior spherical harmonic expansions are usable inside an imaginary, empty sphere. For this application, such a sphere could be positioned in empty space above the intended landing site and rotating with the body. When the spacecraft is inside this sphere, the interior harmonic expansion would be used instead of the conventional, exterior harmonic expansion. Coefficients can be determined by a least-squares fit to gravitation measurements synthesized from conventional formulations. Due to their unfamiliarity, recurrences for interior, as well as exterior, expansions are derived. Hotine's technique for partial derivatives of exterior spherical harmonics is extended to interior harmonics.

Werner, Robert A.↗
NASA NTRS1994

Magnetospheric and ionospheric signals in magnetic observatory monthly means: Electrical conductivity of the deep mantle

First differences of magnetic observatory monthly means for 1963-1982 were analyzed using techniques of spherical harmonics analysis and power spectral analysis. The external source signal is shown to be primarily zonal in geomagnetic coordinates. Prominent peaks are present in the power spectrum at frequencies of 1.0 cycle/yr and 2.0 cycles/yr. The annual signal is largest on the degree 2 external zonal spherical harmonic, while the semiannual signal is largest on the degree 1 and degree 3 external zonal spherical harmonics. The presence of the semiannual signal on odd-degree spherical harmonics and of the annual signal on even-degree spherical harmonics was predicted from symmetry considerations and the annual cycle of solar inclination. These signals are all modulated by the sunspot frequency and its harmonics. The degree 1 term is believed to be due mainly to magnetopause and ring currents while the degree 2 and degree 3 terms are believed to be due mainly to ionospheric currents. The degree 1 external zonal harmonic has a continuous spectrum in addition to the semiannual spectral peak. A corresponding degree 1 internal term is due to electromagnetic induction. The degree 1 continuous spectrum is useful for study of the electrical conductivity of the deep mantle. A global geomagnetic response function consistent with a mantle conductivity of about 10 S/m at the core-mantle boundary has been derived.

Mcleod, Malcolm G.↗
NASA NTRS1981

Equivalent source modeling of the main field using Magsat data

Magsat dipole solution models were obtained with both 32 deg and 21 deg resolution based on a data set extending over 4 months. Time dependence was modeled using first time derivatives for the dipole magnetization vector components. This doubles the n total number of parameters in the solution. The solutions displayed a very slow convergence in time derivatives although at each iteration the conversion of the dipole parameters to spherical harmonic coefficients showed close agreement with the MGST (12/80) spherical harmonic model to degree 13 in the constant terms and 8 in the secular variation terms. The program error in the option to simultaneously estimate observatory anomaly biases was found and corrected. A 32 deg resolution dipole model was generated using the selected magnetic observatory data from 1960-1977 used in the GSFC (9/80) spherical harmonic model.

Source record↗
NASA NTRS1980

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

Equations providing numerical values of the geomagnetic field spherical harmonic spatial power spectrum as defined by Lowes (1966, 1974) are obtained and this power spectrum is related to various other power spectra. Equations relating the spherical harmonic spatial power spectrum to average great circle power spectra for components of the vector magnetic field in the radial direction, along the great circle track and perpendicular to the first two directions are derived under the assumption that the sources of the field are internal. A statistical model for the crustal and core geomagnetic fields is proposed and used to derive equations for the expected main and crustal spherical harmonic power spectra. The model equations are then compared with observations to determine a scale factor which is then used to obtain an estimate for the core radius and a great circle power spectrum for the field component perpendicular to the great circle and radial directions which are in good agreement with observations. The predicted spherical harmonic power spectrum for the crustal field is found to be consistent with POGO satellite and aircraft data. Other possible models for the crustal and core geomagnetic fields are also briefly considered.

Mcleod, M. G.↗
DOE OSTI2023

Scalable self attraction and loading calculations for unstructured ocean tide models

Self attraction and earth-loading effects are important for accurately modeling global tides. A common approach of handling this forcing is to expand mass anomalies into spherical harmonics, which are scaled by load Love numbers to account for elastic earth deformation. We investigate two different approaches to perform these calculations for ocean models that employ unstructured meshes and distributed memory parallelization. The first approach leverages a highly efficient spherical harmonics library, but requires all-to-one and one-to-all communications and interpolation operations between the unstructured and a structured mesh. This approach is compared to a parallel algorithm that computes the spherical harmonic transformations directly on the unstructured mesh with an all-reduce communication. Here, our results show that although the unstructured mesh calculations are more expensive, the scalability of the unstructured mesh approach allows for more efficient spherical harmonics transforms for high-resolution meshes and large processor counts. This methodology enables the efficient inclusion of tidal dynamics large-scale Earth system model simulations.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1972

Sampling functions for geophysics

A set of spherical sampling functions is defined such that they are related to spherical-harmonic functions in the same way that the sampling functions of information theory are related to sine and cosine functions. An orderly distribution of (N + 1) squared sampling points on a sphere is given, for which the (N + 1) squared spherical sampling functions span the same linear manifold as do the spherical-harmonic functions through degree N. The transformations between the spherical sampling functions and the spherical-harmonic functions are given by recurrence relations. The spherical sampling functions of two arguments are extended to three arguments and to nonspherical reference surfaces. Typical applications of this formalism to geophysical topics are sketched.

Giacaglia, G. E. O.↗
DOE OSTI2023

Spatio–Temporal Coarse–Graining Decomposition of the Global Ocean Geostrophic Kinetic Energy

We expand on a recent determination of the first global energy spectrum of the ocean’s surface geostrophic circulation using a coarse-graining (CG) method. We compare spectra from CG to those from spherical harmonics by treating land in a manner consistent with the boundary conditions. While the two methods yield qualitatively consistent domain-averaged results, spherical harmonics spectra are too noisy at gyre-scales (> 1000 km). More importantly, spherical harmonics are inherently global and cannot provide local information connecting scales with currents geographically. CG shows that the extra-tropics mesoscales (100–500 km) have a root-mean-square (rms) velocity of ~15 cm/s, which increases to ~30–40 cm/s locally in the Gulf Stream and Kuroshio and to ~16–28 cm/s in the ACC. There is notable hemispheric asymmetry in mesoscale energy-per-area, which is higher in the north due to continental boundaries. We estimate that ≈25–50% of total geostrophic energy is at scales smaller than 100 km, and is un(der)-resolved by pre-SWOT satellite products. Spectra of the time-mean circulation show that most of its energy (up to 70%) resides in stationary eddies with characteristic scales smaller than (< 500 km). This highlights the preponderance of ‘standing’ small-scale structures in the global ocean due to the temporally coherent forcing by boundaries. By coarse-graining in space and time, we compute the first spatio-temporal global spectrum of geostrophic circulation from AVISO and NEMO. These spectra show that every length-scale evolves over a wide range of time-scales with a consistent peak at ≈200 km and ≈2–3 weeks.

54 ENVIRONMENTAL SCIENCES↗
NASA NTRS1984

Lageos orbit and the albedo problem

The objective was to obtain an analytic expression for the radiation pressure force on a satellite due to sunlight reflected from the Earth. The Lageos satellite undergoes unexplained along-track accelerations. These accelerations are believed to be due mainly to terrestrial radiation pressure. The effect of sunlight reflected off the surface of the Earth must thus be modeled to insure an accurate orbit for Lageos. An accurate orbit is necessary for carrying out Lageos' mission of measuring tectonic plate motion, polar motion, and Earth rotation. The present investigation focuses on a spherical harmonic approach to the problem. An equation for the force was obtained by assuming the Earth's surface reflects sunlight according to Lambert's law. The equation is an integral over the whole Earth's surface. Expressions occurring inside the integral are expressed in terms of spherical harmonics. The problem is thus reduced to integrating products of spherical harmonics.

Rubincam, D. P.↗