The sensitivity of hydraulic head responses to spatially distributed hydraulic parameters is essential for uncertainty analysis, inverse modeling, and parameter estimation and interpretation. This study formulates the Fréchet sensitivity kernel of hydraulic head responses to a suddenly rising boundary and a sinusoidal head fluctuation boundary to variation of spatially distributed hydraulic parameters in a semi-infinite, one-dimensional (1-D), confined aquifer, and it then derives analytical solutions. Different from previous studies that derived expressions for Fréchet kernels in the time domain for a 2-D pumping test, this study is the first to derive the closed-form Fréchet kernels in time and frequency domains for a semi-infinite, 1-D, confined aquifer. This study uses the Fréchet kernels to investigate the nature of singularities in the spatial sensitivity functions around the observation location and boundary. The information content revealed by observation of head change or head fluctuation amplitude at a given specified location and time (or frequency) under the above two boundary conditions is different. When comparing Fréchet sensitivity kernels across various times or periods, multi-frequency information, much like multi-time information, can be instrumental for hydrogeological parameter inversion. The explicit-form Fréchet sensitivity kernels also identify the optimal time or period for obtaining measurements.

中文翻译：

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河流水位上升和潮汐变化边界条件下地下水头对水力参数变化的敏感性比较

水头响应对空间分布水力参数的敏感性对于不确定性分析、反演建模以及参数估计和解释至关重要。本研究制定了水头对突然上升边界和正弦水头波动边界对半无限、一维 (1-D) 承压含水层中空间分布水力参数变化的响应的 Fréchet 敏感核，然后得出解析解。与之前在 2-D 泵浦测试的时域中导出 Fréchet 核表达式的研究不同，本研究首次在时域和频域中导出半无限、一维、承压含水层。本研究使用 Fréchet 核来研究观测位置和边界周围空间灵敏度函数中奇点的性质。在上述两种边界条件下，在给定的指定位置和时间（或频率）下观测水头变化或水头波动幅度所揭示的信息内容是不同的。当比较不同时间或时期的 Fréchet 灵敏度核时，多频率信息，就像多时间信息一样，可以有助于水文地质参数反演。显式形式的 Fréchet 灵敏度内核还确定了获取测量值的最佳时间或周期。