Currently, the least-square estimation method is the mainstream method for recovering spherical harmonic coefficients from area mean values over equiangular blocks. Since the least-square estimation method involves matrix inversion, it requires great computation power when the maximum degree to be solved is large. In comparison, numerical quadrature methods are faster. Recent numerical quadrature methods designed for spherical harmonic analysis of area mean values over blocks delineated by equiangular and Gaussian grids are both fast and exact for band-limited data. However, for band-limited area mean values over an equiangular grid that has \(N\) blocks along the colatitude direction and \(2N\) blocks along the longitude direction, the maximum degree that can be recovered by using current exact numerical quadrature methods is no larger than \(N/2-1\). In this study, by using Lagrange’s method for polynomial interpolation, recently proposed numerical quadrature methods that employ the recurrence relations for the integrals of the associated Legendre’s functions are modified into two new methods. By using these methods, the maximum degree of recovered spherical harmonic coefficients is \(N-1\). The results show that these newly proposed methods are comparable in computation speed with the current numerical quadrature methods and are comparable in accuracy with the least-square estimation method for both band-limited and aliased data. Moreover, solving linear systems is not necessary for these two new methods. The error characteristics of these two new methods are quite different from those of methods that employ least-square methods. The spherical harmonic coefficients recovered using these new methods can effectively supplement those recovered using least-square methods.

目前，最小二乘估计方法是从等角块上的面积平均值中恢复球谐系数的主流方法。由于最小二乘估计方法涉及矩阵求逆，因此当需要求解的最大次数较大时，它需要强大的计算能力。相比之下，数值正交方法更快。最近设计的数值正交方法用于对由等角和高斯网格划定的块上的面积平均值进行球谐分析，对于带宽受限的数据来说既快速又准确。然而，对于沿共纬度方向有 \（N\） 个块和沿经度方向有 \（2N\） 个块的等角网格上的带宽受限面积平均值，使用当前精确数值正交方法可以恢复的最大度数不大于 \（N/2-1\）。在这项研究中，通过使用拉格朗日方法进行多项式插值，最近提出的数值正交方法被修改为两种新方法，该方法对相关勒让德函数的积分采用递归关系。通过使用这些方法，恢复的球谐系数的最大度数为 \（N-1\）。结果表明，这些新提出的方法在计算速度上与目前的数值正交方法相当，对于带限和别名数据，这些方法在准确性上与最小二乘估计方法相当。此外，这两种新方法不需要求解线性系统。这两种新方法的误差特性与采用最小二乘法的方法的误差特性大不相同。 使用这些新方法恢复的球谐系数可以有效地补充使用最小二乘法恢复的球谐系数。