Potential-Field Estimation Using Scalar and Vector Slepian Functions at Satellite Altitude

Abstract

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. The 'altitude-cognizant' gradient vector Slepian functions (AC-GVSF) were first employed in a preceding paper. They enjoy a noise tolerance under downward continuation that is much improved relative to the 'classical' gradient vector Slepian functions (CL-GVSF), which do not factor satellite altitude into their construction. Furthermore, venturing beyond the realm of their first application, in the present article we extend the theory to being able to handle both internal and external potential-field estimation. 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. We explain our solution strategies on the basis of analytic expressions for the behavior of the estimation bias and variance of models for which signal and noise are uncorrelated, (essentially) space-and bandlimited, and spectrally (almost) white. 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.

Ice mass loss in Greenland, the Gulf of Alaska, and the Canadian Archipelago: Seasonal cycles and decadal trends

“…This behavior depends on the properties of the signal itself. The theoretical foundations, statistical implications, and the practicalities of the Slepian methodology are very well understood [ Simons and Plattner , ; Plattner and Simons , ], as indeed they are for various other methods of localization [e.g., Wahr et al , ; Schrama et al , ; Klees et al , ; Velicogna and Wahr , ]. Nevertheless, it is hard to gain a realistic appreciation of the uncertainty of the end products of our analysis without synthetic recovery simulation experiments [e.g., Velicogna and Wahr , ; Harig and Simons , ], which we conducted along the same lines in this paper.…”

Internal and external potential-field estimation from regional vector data at varying satellite altitude

“…(108), namely Eq. (121) is decidedly more palatable when contrasted with the equivalent result (174) of Plattner and Simons (2015b), and thus illustrates the benefits of using altitude-cognizant functions as advocated here. Moreover, we find again eq.…”

“…Our method traces its history to the one-dimensional theory of 'prolate spheroidal wave functions' by Slepian and Pollak (1961), its applications in signal processing (Slepian, 1983), and especially its extensions to scalar spherical fields by and , to spherical vector fields by Plattner and Simons (2014), and to gradient vector spherical functions (curl-free potential fields) by Plattner and Simons (2015b). In the above cited works, satellite altitude, though explicitly considered within the context of the inverse problem, was never a factor in the optimization construction of the Slepian functions, and so we will term them 'canonical' or 'classical'.…”

“…In the above cited works, satellite altitude, though explicitly considered within the context of the inverse problem, was never a factor in the optimization construction of the Slepian functions, and so we will term them 'canonical' or 'classical'. In particular, the functions of Plattner and Simons (2015b) will hereafter be known as 'classical gradient vector Slepian functions' (CL-GVSF). In contrast, the construction by Plattner and Simons (2015a), reformulated in the present paper, does incorporate satellite altitude directly, hence their designation 'altitudecognizant gradient vector Slepian functions' (AC-GVSF).…”

“…Recording gravity and magnetic fields in space is an engineering problem of instrumentation. Mapping such fields from space, back onto the planetary surface where they originate, and separately from any fields generated externally, down to the body of interest, is a problem of inversion (Plattner and Simons, 2015b;Sabaka et al, 2015). Regional modeling is predicated on the ability to include data collected at varying satellite altitude, alleviating noise amplification under 'downward continuation', and, in particular in the case of magnetic field modeling, taking external fields into account.…”