Spherical harmonic expansions are the most popular parametrisation of the gravitational potential and its higher-order spatial derivatives in global geodetic, geophysical, and planetary science applications. The convergence domain of external spherical harmonic expansions is the space above the minimum Brillouin sphere, nevertheless, these series are commonly employed inside this bounding surface without correcting. Justification of this procedure has been debated for several decades, but conclusions among scholars are indefinite and even contradictory.

In this article, we discuss the use of spherical harmonic expansion for the gravitational field modelling inside the minimum Brillouin sphere. In the theoretical part, we systematically summarise the mathematical apparatus of internal and external spherical harmonic series for the gravitational potential, gravitational gradient components, and second-order gravitational tensor components. We also derive analytical downward continuation errors for these quantities in the spectral form. In the experimental part, we evaluate the internal and external spherical harmonic series inside the minimum Brillouin sphere by employing the most recent LOLA topographic and GRAIL gravitational observations of the Moon. We first analyse line-of-sight gravitational accelerations from GRAIL and their forward-modelled counterparts. We next examine seven GRAIL-derived and four forward-modelled global gravitational field models. We further investigate in detail spectral and spatial (signal/error) characteristics from two forward-modelled global gravitational fields – one using the internal and the other employing the external spherical harmonic parametrisation.

Notable findings of this study are: (1) GRAIL measurements taken at low altitudes, below the minimum Brillouin sphere, provide the observational evidence of the divergence in the existing spherical harmonic solutions of the global gravitational field models. (2) Power laws applied at high-frequencies of GRAIL-derived global gravitational field models are too strong. (3) Signal powers of the respective orthogonal components of the gravitational gradient and those of the second-order gravitational tensor are identical above the minimum Brillouin sphere, but may be different inside this bounding surface. (4) Analytical downward continuation of the external spherical harmonic series provides an inhomogeneous gravitational potential spectrum. (5) Vertical–vertical component of the second-order gravitational tensor inside the minimum Brillouin sphere depends on the density and may significantly differ from its equivalent neglecting the density.

Most importantly, we unambiguously confirm that all lunar global gravitational field models based on the external spherical harmonic parametrisation do not correspond to the true counterpart inside the minimum Brillouin sphere. Also, analytically downward continued fields tend to diverge at high frequencies. Therefore, the present gravitational field models of the Moon using external spherical harmonic series must be applied only above the minimum Brillouin sphere (10.2 km above the mean lunar sphere).

The theoretical part represents a rigorous methodological basis for the gravitational field modelling by the internal and external spherical harmonic series. This theory is complete up to the second-order gravitational tensor and holds for any planetary body. The experimental part reveals intricate aspects for the lunar gravitational field. Except for the Moon, however, these practicalities will have substantial implications on future gravitational field determinations of other planetary bodies.

在全球大地测量、地球物理和行星科学应用中，球谐扩展是引力势及其高阶空间导数的最流行的参数化。外部球谐展开的收敛域是最小布里渊球以上的空间，但是，这些级数通常在这个边界面内使用而没有校正。这一程序的合理性已经争论了几十年，但学者之间的结论是不确定的，甚至是相互矛盾的。

在本文中，我们讨论了在最小布里渊球体内的引力场建模中使用球谐展开。在理论部分，我们系统地总结了引力势、引力梯度分量和二阶引力张量分量的内、外球谐级数的数学装置。我们还以频谱形式推导出这些量的解析向下延续误差。在实验部分，我们利用最新的 LOLA 地形和 GRAIL 月球引力观测来评估最小布里渊球内的内部和外部球谐系列。我们首先分析了 GRAIL 及其前向建模对应物的视线重力加速度。我们接下来检查七个 GRAIL 派生的和四个正向建模的全球引力场模型。我们进一步研究了来自两个前向建模的全球引力场的详细光谱和空间（信号/误差）特性——一个使用内部，另一个使用外部球谐参数化。

这项研究的显着发现是： (1) 在最低布里渊球以下的低空进行的 GRAIL 测量提供了全球引力场模型现有球谐解的发散的观测证据。(2) GRAIL 衍生的全球引力场模型高频应用的幂律太强。(3) 引力梯度的各个正交分量和二阶引力张量的信号功率在最小布里渊球以上的信号功率相同，但在该边界面内可能不同。(4) 外球谐系列的解析向下延续提供了一个非均匀的引力势谱。

最重要的是，我们明确确认所有基于外部球谐参数化的月球全球引力场模型并不对应于最小布里渊球内的真实对应物。此外，分析向下的连续场倾向于在高频处发散。因此，目前使用外部球谐系列的月球引力场模型必须仅适用于最小布里渊球 以上（平均月球球以上10.2公里）。

理论部分代表了通过内部和外部球谐系列进行引力场建模的严格方法论基础。这个理论对于二阶引力张量是完整的，并且适用于任何行星体。实验部分揭示了月球引力场的复杂方面。然而，除了月球，这些实用性将对未来其他行星体的引力场测定产生重大影响。