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In this case study, we model a planet's magnetic and gravitational fields using spherical harmonic functions. As an exercise, we analyze data on the Earth's magnetic field collected by NASA's MAGSAT spacecraft, and use it to derive a simple magnetic field model based on these spherical harmonic functions.
Development of least squares process for estimating spherical harmonic coefficients in Mars gravitational potential function based on Viking orbiter spacecraft data
The difficult facing data analysis is the lack of method to handle nonlinear and nonstationary time series. Traditional Fourier-based analyses simply could not be applied here. A new method for analyzing nonlinear and nonstationary data has been developed. The key part is the Empirical Mode Decomposition (EMD) method with which any complicated data set can be decomposed into a finite and often small number of Intrinsic Mode Functions (IMF) that serve as the basis of the representation of the data. This decomposition method is adaptive, and, therefore, highly efficient. The IMFs admit well-behaved Hilbert transforms, and yield instantaneous energy and frequency as functions of time that give sharp identifications of imbedded structures. The final presentation of the results is an energy-frequency-time distribution, designated as the Hilbert Spectrum. Among the main conceptual innovations is the introduction of the instantaneous frequencies for complicated data sets, which eliminate the need of spurious harmonics to represent nonlinear and nonstationary signals. Examples from the numerical results of the classical nonlinear equation systems and data representing natural phenomena are given to demonstrate the power of this new method. The classical nonlinear system data are especially interesting, for they serve to illustrate the roles played by the nonlinear and nonstationary effects in the energy-frequency-time distribution.
The outgoing longwave radiation (OLR) data from ERBE wide-field-of-view (WFOV) and scanning sensor are compared with Nimbus-7 WFOV results. Monthly averaged OLR data from the ERBE WFOV instruments aboard the NOAA-9 and NOAA-10 polar orbiting satellites during the 3-year overlap period with Nimbus-7 are deconvolved using spherical harmonics. Results of a comparison of the data sets are presented on regional, zonal, and global scales in the spatial domain and on a monthly scale in the time domain.
An analysis of the characteristics of the magnetic field of the planet Jupiter is presented. The data were obtained during the flight of Pioneer 11 space probe, using a high field triaxial fluxgate magnetometer. The data are analyzed in terms of traditional Schmitt normalized spherical harmonic expansion fitted to the observations in a least squares sense. Tables of data and graphs are provided to summarize the findings.
Satellite-based observations on river geometries are sporadic in time, space, or both. Most satellite-based surface water maps, river widths, water surface elevations (WSE), slopes, and bathymetry are asynchronized in time and space. The current configuration of satellites such as Sentinel-6 measured the WSE but is missing the river width, slopes, and depths. To advance hydrological sciences research, there is a need to produce a harmonized time series of river geometry data of non-SWOT satellites in partnership with the upcoming SWOT mission. The SWOT satellite will measure river width, height, and slope but missing river depth measurements in space and time. Further, none of these current satellites measure the WSE, river width, and slopes synchronously. In this work, we use the Sentinel-1 SAR satellite data archive from 2015 to the present to create a global river width and surface water database at the reach scale. A modified version of the Sentinel SAR surface water classification algorithm from ASF is used to quantify the surface water extent on the stream approximately every six days (at the equator) at 10m spatial resolution globally. This 10m water mask is fed into a workflow to quantify the river widths, surface water inundations, slopes, and synthetic bathymetry in SWORD (SWOT River Database) stream networks. A Satellite HAND is used to address the cloud obscured surface water observations using a trained machine learning algorithm. We use WSE derived from the Global Water Monitor from NASA GSFC, Hydroweb from LEGOS, and ICESat-2 to harmonize the WSE observation. And Landsat-8/9 and Sentinel-2 water observations to fill the gaps in the Sentinel-1 SAR database. We use Congo River Basin as a test case where we have more than 500 radar altimetry-based WSE, continuous series of Sentinel-1, ICESat-2, Landsat-8/9, and Sentinel-2 observations. A Congo River hydrologic model is used to generate the streamflow discharge. The satellite observed river reaches are assimilated with the stream flows computed by the routing models. And the downstream reaches in the river network without satellite observations get optimized for discharge/river geometry at each observation cycle. Our final product is a harmonized river geometry dataset (reach's water extent, WSE, slope, synthetic bathymetry) for Congo Basin's SWORD reaches.
The field of space physics has a long history of utilizing dimensionality reduction methods to distill data, including but not limited to spherical harmonics, the Fourier Transform, and the wavelet transform. Here, we present a technique for performing dimensionality reduction on ion counts distributions from the Multiscale Mission/Fast Plasma Investigation (MMS/FPI) instrument using a data-adaptive method powered by neural networks. This has applications to both feeding low-dimensional parameterizations of the counts distributions into other machine learning algorithms, and the problem of data compression to reduce transmission volume for space missions. The algorithm presented here is lossy, and in this work, we present the technique of validating the reconstruction performance with calculated plasma moments under the argument that preserving the moments also preserves fluid-level physics, and in turn a degree of scientific validity. The method presented here is an improvement over other lossy compressions in loss-tolerant scenarios like the Multiscale Mission/Fast Plasma Investigation Fast Survey or in non-research space weather applications.
A computer program entitled PAIN (Propeller Aircraft Interior Noise) has been developed to permit calculation of the sound levels in the cabin of a propeller-driven airplane. The fuselage is modeled as a cylinder with a structurally integral floor, the cabin sidewall and floor being stiffened by ring frames, stringers and floor beams of arbitrary configurations. The cabin interior is covered with acoustic treatment and trim. The propeller noise consists of a series of tones at harmonics of the blade passage frequency. Input data required by the program include the mechanical and acoustical properties of the fuselage structure and sidewall trim. Also, the precise propeller noise signature must be defined on a grid that lies in the fuselage skin. The propeller data are generated with a propeller noise prediction program such as the NASA Langley ANOPP program. The program PAIN permits the calculation of the space-average interior sound levels for the first ten harmonics of a propeller rotating alongside the fuselage. User instructions for PAIN are given in the report. Development of the analytical model is presented in NASA CR 3813.
Lageos satellite laser ranging (SLR) data taken over a 2.5 yr period were employed to develop the Goddard Earth Model GEM-L2, a refined gravity field model. Additional data was gathered with 30 other satellites, resulting in spherical harmonics through degree and order 20, based on over 600,000 measurements. The Lageos data was accurate down to 10 cm, after which the GEM 9 data were used to make adjustments past order 7. The resolution of long wavelength activity, through degree and order 4, was made possible by the Lageos data. The GEM-L2 model features a 20 x 20 geopotential, tracking station coordinates (20), 5-day polar motion and A1-UT1 values, and a GM value of 398,600.607 cu km/sq sec. The accuracy of station positioning has been raised to within 6 cm total position globally and within 1.8 cm in baselines. It is concluded that SLR is useful for measuring tectonic plate motions and inter-plate deformations.
An atlas of monthly outgoing longwave radiation global contour maps and associated spherical harmonic coefficients is presented. The atlas contains 23 months of data from November 1985 to October 1987 . The data were derived from the second Earth Radiation Budget (ERB) package, which was flown on the Nimbus 7 Sun-synchronous satellite in 1987. This data set is a companion set and extension to similar atlases that documented 10 years of outgoing longwave radiation results from Nimbus 6 and Nimbus 7 satellites. This atlas and the companion atlases give a data set covering a 12-year time period and will be very useful in studying different aspects of our changing climate. The data set also provides a 3-year overlap with the current Earth Radiation Budget Experiment (ERBE).
Cosmic ray neutron data from the cosmic ray stations from the worldwide network in 1966, 1967 and 1969 are analyzed by means of the three dimensional analysis method by Nagashima. The variations of the north-south anisotropy, which is the first zonal harmonic component obtained from the analysis are studied. The result obtained confirms earlier findings. Relationship of the anisotropy to the interplanetary magnetic field sector polarity is also studied.
Lunar gravitational field obtained from Lunar Orbiter tracking data, with gravity potential as spherical harmonics series and lunar gravitational constant determined
Lunar gravitational field obtained from Lunar Orbiter tracking data, with gravity potential as spherical harmonics series and lunar gravitational constant determined
A strong constraint on 13th order (odd degree) terms in the geopotential has been derived from Navy tracking on a DIADEME 2 fragment (1967-14F). This object (perigee height: 580 km, orbit inclination: 38.9 deg) is presently decaying slowly through perfect commensurability with these terms. The resonance forces will increase its inclination by 0.02 deg when the passage is complete by late 1974. The constraint (lumped harmonics), derived by adjustment of a pair of harmonic coefficients to the Navy inclination data (principally) is: 10 to the 9th power (14.8 + or - 0.8, 48.3 + or - 0.7) = 0.023(C,S)13,13 -0.172(C,S)15,13 0.505(C,S)17,13 - 0.884(C,S)19,13 + (C,S)21,13 0.673(C,S)23,13 0.099(C,S)25,13 0.295(C,S)27,13 -0.279(C,S)29,13 0.018(C,S)31,13 + There should be a significant contribution to this result from terms as high as 29th degree. But current geopotential solutions (for 13th order terms) to this degree are about 20% in error when judged by this independent data.
In situ mass spectrometric measurements of ion and neutral particle thermospheric compositions have been used to infer the latitudinal and diurnal variations of thermospheric atomic hydrogen for solstice conditions. Local time-dependent and local time-independent components of the observed hydrogen distribution were separated on the basis of a model generated by expanding the log of neutral hydrogen concentration in terms of spherical harmonics. Results are compared with analogous data on N2 concentration; the comparison reveals an anticorrelation between gas temperature and hydrogen concentration. The slope of this anticorrelation line represents the 'zero flux condition' from the exosphere theory of Hodges (1973), thus further bearing out the conclusion that exospheric flow is the dominant process governing the global distribution of hydrogen.
Two dimensional gravitational spectra are derived from simple harmonic analysis of range rate tracking data on planetary orbiters. The eccentricity of the orbit is arbitrary and results are shown to vary substantially with the aspect angle of the tracking line of sight with the orbit plane. The development for arbitrary start with stop times (with respect to periapsis) uses modified eccentricity functions evaluated by quadrature. Simulations with a point-masses model of Venus using tracking data on the Pioneer Venus Orbiter show excellent predictions of the average orbiter spectrum over one Venus day. The Venus gravitational signal should be above the tracking noise level for arc lengths longer than 40 deg (in true anomaly) about periapsis and for terms as high as 55th degree. analysis has been made of tracking residuals from a short arc fit to Mariner Mars 9 data over the Hellas Basin (using a complete 6th degree field). Results are most consistent with higher residual gravitational power than predicted from Kaula's rule for Mars.
The steady and unsteady wake profiles of an airfoil with an oscillating flap were measured at nominal free stream Mach number of 0.8 in the NASA Ames 11 x 11-foot wind tunnel. The instantaneous wake velocity and pressure profiles at four axial locations are presented up to one chord length from the trailing edge. Both fundamental harmonic frequency and typical time history data are presented to observe the effects of airfoil incidence and flap angle. The drag coefficient obtained from the wake pressure measurements is compared with that obtained from the airfoil pressure distribution.
Since mid-1978, the medium frequency (2.2 MHz; M.F.) radar at Saskatoon has run continuously, measuring winds by the spaced antenna method in the upper middle atmosphere. As there is so much dynamics activity during the years of MAP it seems very important to establish the features of the tidal field at Saskatoon during 1983 to 1984, and compare these with the climatology from the four years 1978 to 1979 and 1981 to 1982. Wind profiles with 60 to 110 km (80 to 110 km at night) with 3 km resolution are obtained every 5 minutes by means of a full-correlation analysis. Hourly means are formed and harmonic analysis applied to sets of data to obtain the mean winds and tidal oscillation amplitudes and phases. The zonal wind field at Saskatoon is structurally similar to CIRA-72; however, the wind eastward cell is weaker and much more structured, and the summer eastward cell above 85 km is also weaker. For the semidiurnal tide, a strong seasonal change exists, with larger amplitudes and shorter wavelengths, in winter-like months and the reverse in summer-like months. A summary of the tidal parameters at 90 km is given, where mean values from 1978 to 1982 are also shown. This seasonal variation of tides is a midlatitude climatology, as comparisons with Monpazier, France and Christchurch, New Zealand show very similar behavior, as does Durham; and requires antisymmetric tidal modes such as 2,3 and 2,5. Considering the diurnal tide, for most months centered on summer there are long wavelength modes, or superposition of several models. However, some winter months show a short wavelength mode. A summary of the 90 km characteristics is presented.