The problem of optimal linear estimation of functionals depending on the unknown values of a spatial temporal isotropic random field $\zeta(j,x)$, which is periodically correlated with respect to discrete time argument $j\in\mathrm Z$ and mean-square continuous isotropic on the unit sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Estimates are based on observations of the field $\zeta(j,x)+\theta(j,x)$ at points $(j,x):$ $j\in Z\backslash\{0, 1, .... , N\}$, $x\in S_{n}$, where $\theta(j,x)$ is an uncorrelated with $\zeta(t,x)$ spatial temporal isotropic random field, which is periodically correlated with respect to discrete time argument $j\in\mathrm Z$ and mean-square continuous isotropic on the sphere ${S_n}$ with respect to spatial argument $x\in{S_n}$. Formulas for calculating the mean square errors and the spectral characteristics of the optimal linear estimate of the functional are derived in the case where the spectral density matrices are exactly known. Formulas that determine the least favourable spectral density matrices and the minimax (robust) spectral characteristics are proposed in the case where the spectral density matrices are not exactly known but a class of admissible spectral density matrices is given.

翻译：本文研究了依赖于时空各向同性随机场 $\zeta(j,x)$ 未知值之泛函的最优线性估计问题。该随机场关于离散时间参数 $j\in\mathrm Z$ 具有周期相关性，且关于空间参数 $x\in{S_n}$ 在单位球面 ${S_n}$ 上均方连续各向同性。估计基于随机场 $\zeta(j,x)+\theta(j,x)$ 在点集 $(j,x):$ $j\in Z\backslash\{0, 1, .... , N\}$, $x\in S_{n}$ 上的观测值，其中 $\theta(j,x)$ 是与 $\zeta(t,x)$ 不相关的时空各向同性随机场，其同样关于离散时间参数 $j\in\mathrm Z$ 周期相关，且关于空间参数 $x\in{S_n}$ 在球面 ${S_n}$ 上均方连续各向同性。在谱密度矩阵精确已知的情况下，推导了最优线性估计泛函的均方误差计算公式及其谱特征表达式。针对谱密度矩阵未精确已知但给定容许谱密度矩阵类的情形，提出了确定最不利谱密度矩阵及最小最大（鲁棒）谱特征的公式体系。