Spherical harmonic analysis is widely used in all aspects of geoscience. Exact quadrature methods are available for the spherical harmonic analysis of band-limited point values at the grid points of equiangular and Gaussian grids. However, no similarly exact quadrature methods are available for the spherical harmonic analysis of area mean values over the blocks delineated by these grids. In this study, new algorithms appropriate for the exact spherical harmonic analysis of the band-limited area mean values over the blocks delineated by equiangular and Gaussian grids are proposed. For band-limited data, precision that is between that of the least-squares estimation method and of the approximate quadrature methods can be achieved by using the new algorithms. Regarding the computational complexity, fewer operations are needed by the new methods as compared to those needed by the least-squares estimation method and the approximate quadrature methods in the preparation stage when the maximum degree of the spherical harmonic analysis is very large. Simulation experiments are performed to compare the ability to recover the spherical harmonic coefficients by using the least-squares estimation method, the approximate quadrature methods and these new algorithms from aliased data with aliasing components of realistic magnitudes. The results suggest that these new algorithms, with time complexity one order less than that of the least-squares estimation method in the solving stage, perform roughly the same as the least-squares estimation method in recovering spherical harmonic coefficients from the aliased data.

球谐分析广泛应用于地球科学的各个方面。精确的正交方法可用于等角网格和高斯网格的网格点处的带限点值的球谐分析。但是，没有类似的精确正交方法可用于这些网格所描绘的块上的面积平均值的球谐分析。在这项研究中，提出了新的算法，该算法适用于等角网格和高斯网格所描绘的块上的带限区域平均值的精确球谐分析。对于频带受限的数据，可以通过使用新算法来实现介于最小二乘方估计方法和近似正交方法之间的精度。关于计算复杂度，当球形谐波分析的最大程度非常大时，与准备阶段的最小二乘估计方法和近似正交方法相比，新方法所需的操作更少。进行了仿真实验，以比较使用最小二乘估计方法，近似正交方法和这些新算法从具有真实幅度的混叠分量的混叠数据中恢复球形谐波系数的能力。结果表明，这些新算法的时间复杂度比求解阶段的最小二乘估计方法小一个数量级，在从别名数据中恢复球谐系数时，其执行效果与最小二乘估计方法大致相同。