Rectangular harmonic analysis revisited

Abstract

Alldredge's method of rectangular harmonic analysis has been reexamined. After correction of errors, it is found to give improbable values between the data points and wild values outside them. A much more realistic model has been obtained by (1) determining only the most significant coefficients (those that exceed their standard deviations, obtained by an iterative process), (2) introducing new parameters to allow for a linear trend across the region, and (3) increasing the scaling factors so that the sinusoids start and finish outside the region. Modification 1 is the most important for improving the interpolative qualities of the model. Modifications 2 and 3 reduce, but do not entirely eliminate, the wild values near the edges.

Assessment of regional geomagnetic field modelling methods using a standard data set: spherical cap harmonic analysis

“…These results are quite different from those for RHA and RPA, where a large reduction in the number of coefficients was accompanied by only a small increase in rms residual (Malin et al 1996;Düzgit et al 1997).…”

“…After correction of an error, the RHA method of Alldredge (1981) with N max = 5 (43 coefficients) gave 86 nT for the rms residual from the anomaly data (Malin et al 1996). For comparison with SCHA, we here apply the same method to the main field data, with the results given in Table 1.…”

“…The first question we address is the optimum size of cap. Clearly it should be large enough to include all the data, but is there any advantage to be obtained in making it larger, as was found to be the case for RHA boundaries (Malin et al 1996)?…”

“…Alldredge (1981) applied his method of rectangular harmonic analysis (RHA) to a data set based on observatory annual mean values for the European region. The same data were used to assess a modified version of RHA (Malin et al 1996) and rectangular polynomial analysis, RPA (Düzgit et al 1997).…”

Gravity and geoid model in South Korea and its vicinity by spherical cap harmonic analysis

“…To avoid this mathematical inconvenience, Alldredge (1981) introduced a rectangular harmonic analysis (RHA) as a regional base function based on Cartesian coordinates. This method, however, is inappropriate for dealing with potential field data because of large errors near the edges (Malin et al, 1996). Haines (1985) proposed a spherical cap harmonic analysis (SCHA), which gives a uniformly convergent series expansion and poses a basis function including associated Legendre functions of integral order and real number degree (Haines, 1988;Thebault et al, 2006).…”