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Abstract:
Modeling the stochastic behavior of global-scale geophysical signals, such as the Earth’s external gravity field, requires the use of valid (i.e., positive-definite) covariance functions on the sphere. The construction of such functions is still a challenging task, especially when a compactly-supported (i.e., finite) covariance function is needed. A standard approach for deriving a positive definite function is by convolving any function with itself. Accounting for the convolution-multiplication duality, the spectrum of the resulting function is always positive, since it is equal to the multiplication of the spectrum of the self-convolving function with itself. The validity of the resulting function can therefore be easily established. In this study, we utilize this property and develop a new compactly-supported covariance function on the sphere based on the self-convolution of a spherical cap. The shape of this new covariance function resembles other models developed using the self-convolution property. Two representative examples are the spherical covariance function, which is based on the self-convolution of a two-dimensional ball, and the more generalized “Euclid’s hat” function, which is described by the self-convolution of an n-dimensional ball. Although the spatial representation of our newly-developed covariance function is given by a somewhat involved mathematical expression, its spectral representation is rather simple and only requires the calculation of spherical harmonic coefficients of a spherical cap. The latter are also known as Pellinen smoothing coefficients and are commonly evaluated by a well-known recurrence relation for local gravity field modeling applications.
