Fully normalized spherical cap harmonics: application to the analysis of sea-level data from TOPEX/POSEIDON and ERS-1

Abstract

We introduce two sets of fully normalized harmonics for the spectral analysis of functions defined on a spherical cap. The harmonics are the products of Fourier functions and the fully normalized associated Legendre functions of non-integer degree. Using Sturm-Liouville theory for boundary-value problems, we present two convenient and stable formulae for computing the zeros of the associated Legendre functions that form two sets of orthogonal functions. Formulae for the stable numerical evaluation of the fully normalized associated Legendre functions of non-integer degree that avoid the gamma function are also derived. The result from the expansions of sealevel anomaly from altimetry into Set 2 fully normalized cap harmonics shows fast convergence of the series, and the degree variances decay rapidly without aliasing effects. The zero-degree coefficients (Set 2) of sea-level anomaly from TOPEX/ POSEIDON (T/P) and ERS-1 indicate an El Nifio event during 1993 January-1993 July, and a La Niiia event during 1993 November-1994 July, although the ERS-1 result is less obvious. Ocean circulations over the South China Sea and the Kuroshio area are clearly identified with the low-degree expansions of sea-surface topography (SST) from T/P and ERS-1. A cold-core eddy of 4" in diameter centred at 17.5"N, 118"E was detected with the expansion of SST from T/P cycle 47, and a property of the cap harmonics is used to compute this eddy's kinetic energy. The kinetic energy is at a low in winter and high in summer, and its variation seems to be periodic with an amplitude of 0.4 m2 s-'.

Gravity and Topography of the Terrestrial Planets

“…Alternative representations include the use of ellipsoidal harmonics (e.g., Garmier and Barriot, 2001;Jekeli, 1988) when the function is known globally and Slepian functions or spherical cap harmonics (e.g., Haines, 1985;Hwang and Chen, 1997;Thébault et al, 2006Thébault et al, , 2004 when the function is known only regionally. This chapter will use exclusively planetocentric coordinates, where colatitude is measured with respect to the polar axis, but the reader should be aware that many datasets for Earth make use of planetographic coordinates.…”

Modelling of spatial-temporal changes of the geomagnetic field in Japan

“…The SCH modelling is devoted to potential and field representation in a spherical cap. Although the method poses many difficulties (Hwang and Chen, 1997;Thebault et al, 2004), it has been widely employed in deriving regional geomagnetic models, for example, for Canada, Africa and China regions (Haines, 1985;Kotze, 2003;.…”

Observing, Modeling, and Interpreting Magnetic Fields of the Solid Earth

“…The method is claimed to be valid over any spherical cap at any altitude above the Earth's surface. On the basis of these assertions, SCHA has been widely used for producing magnetic anomaly maps (De Santis et al, 1997;Hwang and Chen, 1997;Korte and Haak, 2000).…”