The coefficients of elastic and dissipative operators in a linear hyperbolic SPDE are jointly estimated using multiple spatially localised measurements. As the resolution level of the observations tends to zero, we establish the asymptotic normality of an augmented maximum likelihood estimator. The rate of convergence for the dissipative coefficients matches rates in related parabolic problems, whereas the rate for the elastic parameters also depends on the magnitude of the damping. The analysis of the observed Fisher information matrix relies upon the asymptotic behaviour of rescaled $M, N$-functions generalising the operator sine and cosine families appearing in the undamped wave equation. In contrast to the energetically stable undamped wave equation, the $M, N$-functions emerging within the covariance structure of the local measurements have additional smoothing properties similar to the heat kernel, and their asymptotic behaviour is analysed using functional calculus.

翻译：基于多个空间局部化测量，联合估计线性双曲型随机偏微分方程中弹性算子与耗散算子的系数。随着观测分辨率趋于零，我们建立了增广极大似然估计量的渐近正态性。耗散系数的收敛速率与相关抛物型问题的速率一致，而弹性参数的收敛速率还依赖于阻尼强度。观测费雪信息矩阵的分析依赖于广义化$M, N$函数经尺度变换后的渐近行为，这些函数推广了无阻尼波动方程中出现的算子正弦与余弦族。与能量稳定的无阻尼波动方程不同，局部测量协方差结构中出现的$M, N$函数具有类似热核的附加光滑性质，其渐近行为通过泛函演算进行分析。