The spherical shell and spherical zonal band are two elemental geometries that are often used as benchmarks for gravity field modeling. When applying the spherical shell and spherical zonal band discretized into tesseroids, the errors may be reduced or cancelled for the superposition of the tesseroids due to the spherical symmetry of the spherical shell and spherical zonal band. In previous studies, this superposition error elimination effect (SEEE) of the spherical shell and spherical zonal band has not been taken seriously, and it needs to be investigated carefully. In this contribution, the analytical formulas of the signal of derivatives of the gravitational potential up to third order (e.g., V, \(V_{z}\), \(V_{zz}\), \(V_{xx}\), \(V_{yy}\), \(V_{zzz}\), \(V_{xxz}\), and \(V_{yyz}\)) of a tesseroid are derived when the computation point is situated on the polar axis. In comparison with prior research, simpler analytical expressions of the gravitational effects of a spherical zonal band are derived from these novel expressions of a tesseroid. In the numerical experiments, the relative errors of the gravitational effects of the individual tesseroid are compared to those of the spherical zonal band and spherical shell not only with different 3D Gauss–Legendre quadrature orders ranging from (1,1,1) to (7,7,7) but also with different grid sizes (i.e., \(5^{\circ }\times 5^{\circ }\), \(2^{\circ }\times 2^{\circ }\), \(1^{\circ }\times 1^{\circ }\), \(30^{\prime }\times 30^{\prime }\), and \(15^{\prime }\times 15^{\prime }\)) at a satellite altitude of 260 km. Numerical results reveal that the SEEE does not occur for the gravitational components V, \(V_{z}\), \(V_{zz}\), and \(V_{zzz}\) of a spherical zonal band discretized into tesseroids. The SEEE can be found for the \(V_{xx}\) and \(V_{yy}\), whereas the superposition error effect exists for the \(V_{xxz}\) and \(V_{yyz}\) of a spherical zonal band discretized into tesseroids on the overall average. In most instances, the SEEE occurs for a spherical shell discretized into tesseroids. In summary, numerical experiments demonstrate the existence of the SEEE of a spherical zonal band and a spherical shell, and the analytical solutions for a tesseroid can benefit the investigation of the SEEE. The single tesseroid benchmark can be proposed in comparison to the spherical shell and spherical zonal band benchmarks in gravity field modeling based on these new analytical formulas of a tesseroid.

球壳和球带是两个基本几何形状，通常用作重力场建模的基准。当将球壳和球状带状带离散化为细小体时，由于球壳和球状带状带的球对称性，可以减少或抵消细小体叠加的误差。在以往的研究中，球壳和球带的这种叠加误差消除效应（SEEE）并未引起重视，需要仔细研究。在这个贡献中，引力势导数信号的解析公式高达三阶（例如，V，\（V_{z}\），\（V_{zz}\），\（V_{xx}\ )当计算点位于_ _ _ _ _极轴。与先前的研究相比，球形带状带的引力效应的更简单的解析表达式是从这些新的 tesseroid 表达式中导出的。在数值实验中，单个 tesseroid 的引力效应的相对误差与球形带状带和球壳的引力效应进行了比较，不仅具有从 (1,1,1) 到 (7) 的不同 3D Gauss–Legendre 正交阶数,7,7) 但也有不同的网格大小（即\(5^{\circ }\times 5^{\circ }\) , \(2^{\circ }\times 2^{\circ }\ ), \(1^{\circ }\times 1^{\circ }\) , \(30^{\prime }\times 30^{\prime }\)和\(15^{\prime }\times 15^{\prime }\) ) 在 260 公里的卫星高度。数值结果表明，SEEE 不会发生在离散为 tesseroids 的球形带状带的引力分量V，\(V_{z}\)，\(V_{zz}\)和\(V_{zzz}\) . \(V_{xx}\)和\(V_{yy}\)可以找到 SEEE ，而\(V_{xxz}\)和\(V_{yyz}\)存在叠加误差效应球形带状带的整体平均离散化为 tesseroids。在大多数情况下，SEEE 出现在离散为 tesseroids 的球壳上。综上所述，数值实验证明了球形带状带和球壳的 SEEE 的存在，而 tesseroid 的解析解有助于 SEEE 的研究。与基于这些新的 tesseroid 分析公式的重力场建模中的球壳和球形带基准相比，可以提出单个 tesseroid 基准。