Analytical solutions were derived for the variance, covariance, and spectrum of
groundwater level,

Groundwater level or hydraulic head (

It is obvious that errors always exist in the groundwater levels calculated or simulated with analytical or numerical solutions. The main sources of errors include the simplification or approximation in a conceptual model and uncertainties in the model parameters. Problems in conceptualization or model structure have been dealt with by many researchers (Neuman, 2003; Rojas et al., 2008, 2010; Ye et al., 2008; Refsgaard et al., 2007; Zeng et al., 2013). Uncertainties in the model parameters (e.g., hydraulic conductivity, recharge rate, evapotranspiration, and river conductance) have been investigated, based on generalized likelihood uncertainty estimation and Bayesian methods (Nowak et al., 2010; Neuman et al., 2012; Rojas et al., 2008, 2010). The uncertainty in groundwater level has been one of the main research topics in stochastic subsurface hydrology for more than 3 decades. Most of these studies were focused on the spatial variability of groundwater level due to aquifers' heterogeneity (Dagan, 1989; Gelhar, 1993; Zhang, 2002). Little attention has been given to the uncertainties in groundwater level due to temporal variations in hydrological processes, e.g., recharge, evapotranspiration, discharge to a river, and river stage (Bloomfield and Little, 2010; Zhang and Schilling, 2004; Schilling and Zhang, 2012; Liang and Zhang, 2013a; Zhu et al., 2012).

Uncertainties in groundwater level fluctuations have been studied by Zhang
and Li (2005, 2006) and most recently by Liang and Zhang (2013a).
Based on a linear reservoir model with a white noise or
temporally correlated recharge process, Zhang and Li (2005, 2006) derived
the variance and covariance of

In this study we extended the above-mentioned work by considering the
groundwater flow in a bounded aquifer, described by a linearized Boussinesq
equation, with a random source/sink as well as random initial and boundary
conditions, since the latter processes are known to give uncertainties. The
objectives of this study are (1) to derive analytical solutions for the
covariance, variance, and spectrum of groundwater level, and (2) to
investigate the individual and combined effects of these random processes on
uncertainties and scaling of

Under the Dupuit assumption, the one-dimensional transient groundwater flow
in an unconfined aquifer near a river (Fig. 1) can be approximated with the
linearized Boussinesq equation (Bear, 1972) with the initial and
boundary conditions, i.e.,

A schematic of the unconfined aquifer studied, where

The groundwater level,

Following Gelhar (1993, p. 34), we express the spectra of

The spectral density of

The general expression of the head variance in Eq. (8) depends on the
variances of the four random processes,

The graphs on the left column show the standard deviation
(

We first evaluate the effect of the random initial condition due to the
random term,

We then consider the uncertainty in the areal source/sink term (

Thirdly, we investigate the effect of the left random flux boundary by
setting

Fourthly, we investigated the effect of the random head boundary by setting

Finally, we consider the combined effects of the uncertainties from all four
sources, i.e., the initial condition, sources, and flux and head boundaries.
The head variance in Eq. (8) is written in the dimensionless form as

The changes of this

The dimensionless power spectrum versus frequency (

We first evaluated

Secondly, the spectrum

Thirdly, the spectrum

Finally, the head spectrum due to the combined effect of all three random
sources (the white noise recharge, and flux and head boundaries) was
evaluated, i.e.,

In this study the effects of random source/sink, and initial and boundary
conditions on the uncertainty and temporal scaling of the groundwater level,

The error in

The error in

The errors in

In the typical sandy aquifer studied (with the length of aquifer at the
direction of water flow

The aquifer system behaves as a low-pass filter which filters the short-term (high frequencies) fluctuations and keeps the long-term (low frequencies) fluctuations.

Temporal scaling of groundwater level fluctuations may indeed exist 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.

This study was partially supported with research grants from the National Nature Science Foundation of China (NSFC-41272260; NSFC-41330314; NSFC-41302180), the Natural Science Foundation of Jiangsu Province (SBK201341336) and from the National Key project “Water Pollution Control in the Huai River Basin” of China (2012ZX07204-001, 2012ZX07204-003). Edited by: S. Attinger