This work develops a transfer function to describe the variation in the integrated specific discharge in response to the temporal variation in the rainfall event in the frequency domain. It is assumed that the rainfall–discharge process takes place in a confined aquifer with variable thickness, and it is treated as nonstationary in time to represent the stochastic nature of the hydrological process. The presented transfer function can be used to quantify the variability in the integrated discharge field induced by the variation in rainfall field or to simulate the discharge response of the system to any varying rainfall input at any time resolution using the convolution model. It is shown that, with the Fourier–Stieltjes representation approach, a closed-form expression for the transfer function in the frequency domain can be obtained, which provides a basis for the analysis of the influence of controlling parameters occurring in the rainfall rate and integrated discharge models on the transfer function.

Quantifying the variability in the specific discharge response of an aquifer system to fluctuations in inflow recharge is essential for efficient groundwater resources management. However, this requires extensive and continuous hydrological time series data, and these data are very often not available in practice. A possible approach (namely convolution or transfer function approach) to this problem is to simulate the discharge response by convoluting the time-varying recharge input with the corresponding impulse response. In convolution models, the aquifer is regarded as a filter that converts recharge signals into fluctuations of the aquifer head or discharge. Lumped conceptual–convolution models have been shown to be an efficient means for the simulation of the time series of groundwater levels (e.g., Gelhar, 1974; Molénat et al., 1999; Olsthoorn, 2007; Long and Mahler, 2013; Pedretti et al., 2016).

Since the impulse response function in the convolution model contains all information of the system necessary to relate its input to its output, it may be determined from the analytical solution of the linear system equation governing the input–output process (e.g., Cooper and Rorabaugh, 1963). Once a suitable impulse response function can be specified, it allows the simulation of the linear system response to any varying input at any time resolution.

In this work, a regional-scale flow in a confined aquifer with variable thickness, which is recharged by rainfall through an outcrop, is analyzed by deriving transfer functions to characterize the rainfall–discharge process in the frequency domain. The stochastic analysis of groundwater flow is traditionally based on the assumption of the stationarity of the recharge and discharge processes. However, the hydrologic process in nature is nonstationary stochastic (e.g., Christensen and Lettenmaier, 2007; Milly et al., 2008; Sang et al., 2018). In order to improve the quantification of the natural recharge–discharge process, the nonstationary rainfall–discharge process is assumed in this study. The Fourier–Stieltjes representation approach is used to achieve the goal of this work. The analysis of the results is focused on the influence of controlling parameters in the rainfall–discharge models on the transfer function.

In certain areas, aquifer recharge can vary greatly over time, so determining the discharge of the aquifer at the outlet for regional groundwater problems, which involves transferring recharge at the aquifer outcrop over a relatively large space scale, can be quite difficult. However, it is very important for the planning and management of regional groundwater resources that require knowledge of discharge at the aquifer outlet over a long period of time. This study is, therefore, devoted to quantifying the discharge response of the confined aquifer at the outlet to the temporal variation in aquifer recharge.

In this study, a confined aquifer with variable thickness is considered as a
linear block box system, with a stochastic rainfall recharge input and,
therefore, a stochastic runoff output. Both inputs and outputs are variable
in time. In a linear system, the output of the system can be represented as
a linear combination of the responses to each of the basic inputs through
the convolution integral on a continuous timescale as (e.g., Rugh, 1981;
Rinaldo and Marani, 1987) as follows:

Schematic representation of a linear block box system.

When using the nonstationary Fourier–Stieltjes representations for the
perturbed quantities of random recharge and outflow discharge processes,
namely (e.g., Priestley, 1965) in the following:

In practice, the interest in many cases resides in evaluating the influence of the variation in the recharge on the variation in the outflow discharge. Equation (4) provides an efficient way of quantifying the variability in the outflow induced by the fluctuations in the inflow process in the frequency domain, since it relates the fluctuations in an output time series to those of an input series.

It is worthwhile mentioning that, for the case of second-order stationary
rainfall processes, the representations of the forms (2) and (3) are
reduced, respectively, to the following:

In the following, the focus is on the development of a closed-form expression for the transfer function for a linear lumped confined flow model in which the regional confined aquifer is directly recharged by rainfall in the area corresponding to the high-elevation outcrop.

The differential equation describing the transient flow of groundwater in
inhomogeneous isotropic confined aquifers is of the following form (e.g., Bear, 1979; de Marsily, 1986):

Integrating Eq. (10) along the

The use of the depth-averaged hydraulic head operator for modeling regional groundwater flow is valid when the variation in aquifer thickness is much smaller than the average thickness (Bear, 1979; Bear and Cheng, 2010). The error introduced by the use of this operator is very small in most cases of practical interest, greatly simplifying the analysis of flow in confined aquifers.

Similarly, when applying Leibniz's rule to the Darcy equation, the vertically
integrated specific discharge in the

In this study, the regional confined aquifer is considered with a
nonuniform, unidirectional mean flow in the

In the following analysis, the recharge rate is considered a random function
of time. Equation (15) is then regarded as a stochastic differential
equation with a stochastic input in time and, therefore, a stochastic output
in time. Introduction of the decomposition of the depth-averaged hydraulic head
into a mean and zero-mean perturbation into Eq. (16), after
subtracting the mean of the resulting equation from Eq. (16), means that the result is the following equation describing the depth-averaged head perturbation:

If it is assumed that the thickness of the confined aquifer increases
exponentially in the

The following Fourier–Stieltjes integral representation of a depth-averaged
head process is used to solve Eqs. (19) and (20) for the fluctuations

By solving the above boundary value problem, the oscillatory function of
depth-averaged head process is found to be the following (see Appendix A):

Equation (25) implies that the transfer function

Furthermore, the rest of this study takes into account that rain falls
within a defined period of time over a certain area of horizontal extension
from

Using the Fourier–Stieltjes integral representation for the
perturbation

In the case where the regional confined aquifer is directly recharged by
rainfall at the aquifer outcrop (

The transfer function of the rainfall processes in Eq. (35) behaves like a
filter, attenuating the high-frequency part of the rainfall spectrum. The
graph of the transfer function, which is characterized by the characteristic
timescale

Graphical representation of the transfer function of the rainfall
processes in Eq. (35) characterized by the timescale for different length
scales, where the series calculation is truncated up to

Through the use of Eqs. (25) and (34), the oscillatory function of the
integrated discharge process could be represented as follows:

An essential feature of the transfer function of the integrated discharge flux in Eq. (37) is the resulting filtering associated with the flow process, as shown in Fig. 3. The attenuation of the high-frequency part of the flow discharge spectrum means that the flow process smooths out much of the small-scale variations caused by the rainfall field. Physically, this feature implies that the flow field is much smoother than the rainfall field. The figure also shows that the transfer function at fixed values for frequency and time increases with the increasing thickness of the confined aquifer. An increase in the thickness of the aquifer leads to an increased temporal persistence of the flow discharge fluctuations caused by the variation in the rainfall field and, thus, to an increase in the variability in integrated discharge field. As shown in Fig. 4, the ratio of the mean hydraulic conductivity to the storage coefficient (often referred to as the aquifer diffusivity) plays a similar role in influencing the variation in the transfer function to the thickness of the confined aquifer. The introduction of a larger aquifer diffusivity leads to a larger transfer function of integrated discharge and, thus, to a larger variability in the discharge field. Since the variability in the discharge field is positively correlated with that of rainfall field, the variability in the integrated discharge field will decrease with the increasing characteristic timescale or length scale of the rainfall field (see Fig. 2).

Influence of the thickness of the confined aquifer on the transfer
function of the discharge flux, where the series calculation is truncated up
to

Influence of the aquifer diffusivity on the transfer function of
the discharge flux, where the series calculation is truncated up to

From Eqs. (4) or (8), the transfer function can be defined as the ratio of the fluctuations of an observation of output time series to those of input time series in a frequency domain. Equations (35) and (37) indicate that the transfer functions are related to the properties of the rainfall field and the aquifer, such as the characteristic scales of the time and length of the rainfall field and the diffusivity and thickness parameters of the aquifer. Therefore, the transfer function derived here has the potential to perform a parameter estimation based on the observations of input and output time series using the inverse modeling approach.

The traditional approach to regional groundwater flow problems introduces
the transmissivity term, namely the depth-integrated hydraulic conductivity
operator, as follows:

This means that the effects of both the variation in

The proposed diffusion equation of this work is the following:

Climate changes have a direct influence on the rainfall event (e.g., Trenberth, 2011; Pendergrass et al., 2014; Eekhout et al., 2018). The nonstationarity in the statistical properties of a rainfall field is a representation of climate change (e.g., Razavi et al., 2015; López and Francés, 2013; Benoit et al., 2020). The nonstationary effect of climatic change over time on variability in groundwater-specific discharge has not yet been well characterized. The transfer function in Eq. (37), which relates the nonstationary spectra of the rainfall fluctuations to those of integrated discharge variation, generalizes existing studies that considered stationary recharge and discharge fields. To our knowledge, it has not been previously presented in the literature and has the potential for analyzing the effects of climate change on temporal groundwater-specific discharge variability.

The usefulness of the stochastic theory presented here lies in its
essentially predictive nature. The variance can be used as a quantification
of the uncertainty associated with the prediction in field situations using
the linear system model. In this sense, the solution of Eq. (1)

For large times, the first term in Eq. (37) dominates the sum of the other
terms, and therefore, the transfer function can be approximated by the following:

After observing the recharge rate

If the temporal random rainfall fields are stationary, there exists a
representation of the rainfall perturbation process in terms of a
Fourier–Stieltjes integral in the form of Eq. (6). Substituting Eqs. (6) and (21) into Eq. (19) gives the following:

At large timescales, Eq. (35) approaches a finite value as follows:

An analytical transfer function is developed to describe the spectral response characteristics of confined aquifers with variable thickness to the variation in the rainfall field, where the aquifer is directly recharged by rainfall at the outcrop of the aquifer. The rainfall–discharge process is treated as nonstationary in time, as it reflects the stochastic nature of the hydrological process. Any varying rainfall input at any time resolution can be convolved with the transfer function (or impulse response function) to simulate any discharge output of a linear model. The transfer function derived here, which relates the nonstationary spectra of the rainfall fluctuations to those of integrated discharge variation, has the potential to analyze the influence of climate change on groundwater recharge variability.

The closed-form results of this work are developed on the basis of the
Fourier–Stieltjes representation approach, which allows us to analyze the
effects of the controlling parameters in the models on the transfer function
of the integrated discharge. It is found that the persistence in rainfall
fluctuations is greater for a smaller value of the characteristic timescale or
length scale of the rainfall field, which, in turn, leads to greater
variability in the integrated discharge field. The attenuating
characteristic of the confined aquifer flow system is observed in the
spectral domain. The variability in the integrated discharge in a confined
aquifer with variable thickness is increased with the thickness
parameter

The boundary value problem describing the depth-averaged head fluctuations
induced by the variation in recharge rate in frequency domain is given by
Eqs. (22) and (23). Using the transformation, as follows:

Equation (22) in

The solution of Eqs. (A2) and (A3) can be found through the technique of the
separation of variables (e.g., Farlow, 1993) as follows:

Making use of the following transformation:

In a similar way, based on the technique of the separation of variables, Eqs. (B2) and (B3) arrive at the solution in the following form:

No data sets were used in this article.

CMC, CFN, WCL, CPL and IHL conceptualized the paper, developed the methodology, conducted the formal analysis, prepared the original draft and reviewed and edited the paper. CFN also supervised the project and acquired the funding.

The authors declare that they have no conflict of interest.

Research leading to this paper has been partially supported by the Taiwan Ministry of Science and Technology (grant nos. MOST 108-2638-E-008-001-MY2, MOST 108-2625-M-008-007, and MOST 107-2116-M-008-003-MY2. We are grateful to the editor, Nadia Ursino, and the anonymous referees for the constructive comments that improved the quality of this work.

This research has been supported by the Ministry of Science and Technology, Taiwan (grant nos. MOST 108-2638-E-008-001-MY2, MOST 108-2625-M-008-007, and MOST 107-2116-M-008-003-MY2).

This paper was edited by Nadia Ursino and reviewed by two anonymous referees.