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Random fields on the sphere play a fundamental role in the natural sciences.This paper presents a simulation algorithm parenthetical to the spectralturning bands method used in Euclidean spaces, for simulating scalar- orvector-valued Gaussian random fields on the $d$-dimensional unit sphere. Thesimulated random field is obtained by a sum of Gegenbauer waves, each of whichis variable along a randomly oriented arc and constant along the parallelsorthogonal to the arc. Convergence criteria based on the Berry-Ess\'eeninequality are proposed to choose suitable parameters for the implementation ofthe algorithm, which is illustrated through numerical experiments. A by-productof this work is a closed-form expression of the Schoenberg coefficientsassociated with the Chentsov and exponential covariance models on spheres ofdimensions greater than or equal to 2.

The Turning Arcs: a Computationally Efficient Algorithm to Simulate Isotropic Vector-Valued Gaussian Random Fields on the $d$-Sphere