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Analytical solutions were derived for the variance, covariance, and spectrum of groundwater level, <i>h</i>(<i>x</i>, <i>t</i>), in an unconfined aquifer described by a linearized Boussinesq equation, with random source/sink and initial and boundary conditions. It was found that in a typical aquifer, the error in <i>h</i>(<i>x</i>, <i>t</i>) at an early point in time is mainly caused by the random initial condition, and the error reduces as time progresses to reach a constant error at a later time. The duration for which the effect of the random initial condition is significant may be a few hundred days in most aquifers. The constant error in <i>h</i>(<i>x</i>, <i>t</i>) at a later time is due to the combined effects of the uncertainties in the source/sink and flux boundary: the closer to the flux boundary, the larger the error. The error caused by the uncertain head boundary is limited to a narrow zone near the boundary and remains more or less constant over time. The aquifer system behaves as a low-pass filter which filters out high-frequency noises and keeps low-frequency variations. Temporal scaling of groundwater level fluctuations exists in most parts of a low permeable aquifer whose horizontal length is much larger than its thickness, caused by the temporal fluctuations of areal source/sink.