20 research outputs found
Internal and external potential-field estimation from regional vector data at varying satellite altitude
When modeling global satellite data to recover a planetary magnetic or
gravitational potential field and evaluate it elsewhere, the method of choice
remains their analysis in terms of spherical harmonics. When only regional data
are available, or when data quality varies strongly with geographic location,
the inversion problem becomes severely ill-posed. In those cases, adopting
explicitly local methods is to be preferred over adapting global ones (e.g., by
regularization). Here, we develop the theory behind a procedure to invert for
planetary potential fields from vector observations collected within a
spatially bounded region at varying satellite altitude. Our method relies on
the construction of spatiospectrally localized bases of functions that mitigate
the noise amplification caused by downward continuation (from the satellite
altitude to the planetary surface) while balancing the conflicting demands for
spatial concentration and spectral limitation. Solving simultaneously for
internal and external fields in the same setting of regional data availability
reduces internal-field artifacts introduced by downward-continuing unmodeled
external fields, as we show with numerical examples. The AC-GVSF are optimal
linear combinations of vector spherical harmonics. Their construction is not
altogether very computationally demanding when the concentration domains (the
regions of spatial concentration) have circular symmetry, e.g., on spherical
caps or rings - even when the spherical-harmonic bandwidth is large. Data
inversion proceeds by solving for the expansion coefficients of truncated
function sequences, by least-squares analysis in a reduced-dimensional space.
Hence, our method brings high-resolution regional potential-field modeling from
incomplete and noisy vector-valued satellite data within reach of contemporary
desktop machines.Comment: Under revision for Geophys. J. Int. Supported by NASA grant
NNX14AM29
Spatiospectral concentration of vector fields on a sphere
We construct spherical vector bases that are bandlimited and spatially
concentrated, or, alternatively, spacelimited and spectrally concentrated,
suitable for the analysis and representation of real-valued vector fields on
the surface of the unit sphere, as arises in the natural and biomedical
sciences, and engineering. Building on the original approach of Slepian,
Landau, and Pollak we concentrate the energy of our function bases into
arbitrarily shaped regions of interest on the sphere, and within certain
bandlimits in the vector spherical-harmonic domain. As with the concentration
problem for scalar functions on the sphere, which has been treated in detail
elsewhere, a Slepian vector basis can be constructed by solving a
finite-dimensional algebraic eigenvalue problem. The eigenvalue problem
decouples into separate problems for the radial and tangential components. For
regions with advanced symmetry such as polar caps, the spectral concentration
kernel matrix is very easily calculated and block-diagonal, lending itself to
efficient diagonalization. The number of spatiospectrally well-concentrated
vector fields is well estimated by a Shannon number that only depends on the
area of the target region and the maximal spherical-harmonic degree or
bandwidth. The spherical Slepian vector basis is doubly orthogonal, both over
the entire sphere and over the geographic target region. Like its scalar
counterparts it should be a powerful tool in the inversion, approximation and
extension of bandlimited fields on the sphere: vector fields such as gravity
and magnetism in the earth and planetary sciences, or electromagnetic fields in
optics, antenna theory and medical imaging.Comment: Submitted to Applied and Computational Harmonic Analysi
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High-resolution local magnetic field models for the Martian South Pole from Mars Global Surveyor data
We present two high‐resolution local models for the crustal magnetic field of the Martian south polar region. Models SP130 and SP130M were derived from three‐component measurements made by Mars Global Surveyor at nighttime and at low altitude (<200 km). The availability area for these data covers the annulus between latitudes −76° and −87° and contains a strongly magnetized region (southern parts of Terra Sirenum) adjacent to weakly magnetized terrains (such as Prometheus Planum). Our localized field inversions take into account the region of data availability, a finite spectral bandlimit (spherical harmonic degree L = 130), and the varying satellite altitude at each observation point. We downward continue the local field solutions to a sphere of Martian polar radius 3376 km. While weakly magnetized areas in model SP130 contain inversion artifacts caused by strongly magnetized crust nearby, these artifacts are largely avoided in model SP130M, a mosaic of inversion results obtained by independently solving for the fields over individual subregions. Robust features of both models are magnetic stripes of alternating polarity in southern Terra Sirenum that end abruptly at the rim of Prometheus Planum, an impact crater with a weak or undetectable magnetic field. From a prominent and isolated dipole‐like magnetic feature close to Australe Montes, we estimate a paleopole with a best fit location at longitude 207° and latitude 48°. From the abruptly ending magnetic field stripes, we estimate average magnetization values of up to 15 A/m
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A spatiospectral localization approach for analyzing and representing vector-valued functions on spherical surfaces
We review the construction of three different Slepian bases on the sphere, and illustrate their theoretical behavior and practical use for solving ill-posed satellite inverse problems. The first basis is scalar, the second vectorial, and the third suitable for the vector representation of the harmonic potential fields on which we focus our analysis. When data are noisy and incompletely observed over contiguous domains covering parts of the sphere at satellite altitude, expanding the unknown solution in terms of a Slepian basis and seeking truncated expansions to achieve least-squares data fit has advantages over conventional approaches that include the ease with which the solutions can be computed, and a clear statistical understanding of the competing effects of solution bias and variance in modulating the mean squared error, as we illustrate with several new examples
csdms-contrib/slepian_hotel: Cascade
First version of vector Slepian functions applications for planetary potential field inversion
3-D electrical resistivity tomography using adaptive wavelet parameter grids
ISSN:0956-540XISSN:1365-246