The influence of the temporal changes in lateral inflow rate on the discharge variability in stream channels is explored through the analysis of the diffusion wave equation (i.e. the linearized Saint-Venant equation). To account for variability and uncertainty, the lateral inflow rate is regarded as a temporal random function. On the basis of the spectral representation theory, analytical expressions for the covariance function and evolutionary power spectral density of the random discharge perturbation process are derived to quantify variability in stream flow discharge induced by the temporal changes in lateral inflow rate. The treatment of the discharge variance (square root of the variance) gives us a quantitative estimate of uncertainty in predictions from the deterministic model. It is found that the discharge variability of stream flow is very large in the downstream reach, indicating large uncertainty anticipated from the use of the deterministic model. A larger temporal correlation scale of inflow rate fluctuations, representing more temporal consistency of fluctuations in inflow rate around the mean, introduces a higher variability in stream flow discharge.

Surface runoff originates from precipitation intensities exceeding the infiltration capacity of the surface (e.g. Duan et al., 1992; Sivakumar et al., 2000; Ruiz-Villanueva et al., 2012; Valipour, 2015). This process may result in lateral inflow to nearby stream channels. Significant lateral inflows may contribute to streams during storm-runoff periods when stream reaches are of large lateral watershed areas or upslope accumulated areas (Jencso et al., 2009). These lateral inflows may be not only a source of water to streams, but also a source of contaminants to surface water. Agricultural chemicals are frequently mixed into shallow soil layers and lateral inflows may cause the release and migration of them into streams (Govindaraju, 1996). The effect of the lateral inflow on the stream flow provides an important basis for analyzing contaminant transport in surface water. Understanding and quantification of the influence of inflow process on stream flow discharge is therefore essential for water resource planning and management.

Natural variability, such as significant variability of rainfall events on both temporal and spatial scales (e.g. Ogden and Julien, 1993; Redano and Lorente, 1993; Wheater et al., 2000; Zhang et al., 2001; De Michele and Bernardara, 2005; Haberlandt et al., 2008; Valipour, 2012; Bewket and Lal, 2014) and the great heterogeneity of soil types at the ground surface (e.g. Jencso et al., 2009; Fournier et al., 2013) and surface saturation (e.g. Schumann et al., 2009; Riley and Shen, 2014) over a watershed, creates a very complex runoff process on the land surface. Many practical problems of flood wave routing require predictions over relatively large time and space scales. The key issue is how one can realistically incorporate the effect of natural heterogeneity into models to predict flood wave behavior at large time and space scales. Due to a high degree of the natural heterogeneity of the surface runoff process, the use of deterministic analysis techniques in stream flow modeling is inevitably subject to large uncertainty. The theoretical understanding of variability in flood wave routing is far from complete. Motivated by that, this article focuses on quantification of the discharge variability in a lateral-inflow-dominated stream.

In the following, the response of the transient stream flow process to spatiotemporal lateral inflow in a diffusion wave model is analyzed stochastically by treating the fluctuations in lateral inflow rate as temporal stationary random processes. The non-stationary spectral techniques are employed to obtain closed-form solutions for quantifying the discharge variability in stream channels. These solutions provide variance relations for flow discharge, and thereby allow for assessing the impact of statistical properties of lateral inflow rate process on the discharge variability.

To the best of our knowledge, the issue on quantifying the effect of temporal variation of lateral inflow on the stream flow variability using non-stationary spectral techniques so far has not been addressed. The approach presented herein provides not only an analytical methodology but also a basic framework for understanding the response of transient stream flow process and quantifying the stream flow variability. It is hoped that the proposed approach and our findings obtained in this study are useful for further research in this area.

This study considers the case of unsteady flow in open channels. The
equations that describe the propagation of a flood wave with respect to
distance along the channel and time in open channels are the so-called
Saint-Venant equations, consisting of the continuity equation and the
momentum equation. For most flood events, in most rivers the inertial terms
appearing in the momentum equation of the Saint-Venant equations can be
neglected as they are relatively smaller than the terms arising from gravity
and resistance forces (Henderson, 1963; Dooge and Harley, 1967; Daluz Viera,
1983), leading to a simplified model of open channel flow. The diffusion
wave equation is then expressed as (e.g. Moussa, 1996; Sivapalan et al., 1997)

Equation (1) is a nonlinear partial differential equation and has a complex
behavior of the stream flow in general. No analytical solution of Eq. (1) is
available in the literature. However, the problem can be solved analytically
by some simplifications to Eq. (1), such as linearization for the case of an
initially steady uniform flow. On the basis of expansion of the dependent
variable and the nonlinear terms in Eq. (1) around the initial condition of
steady uniform flow and limitation of the expansion to the first-order
variation from the steady state, the resulting linearized Eq. (1) can be written as

The problem of interest here is the stream flow response to the temporal changes in lateral inflow rate, which is governed by Eq. (2). The solution to Eq. (2) with associated initial and boundary conditions will serve as the starting point for conducting the following investigation of stream flow variability.

To derive the analytical solution of Eq. (2), one needs to specify the form
of

Consider that the flow domain is bounded within the range 0

The approach followed is to develop the analytical solution of Eq. (2) in the Fourier frequency domain.

Temporal stationarity of the

It follows from Eqs. (5) and (7) that Eq. (2) takes the form

The infinite series in Eq. (10) converges rapidly when

The use of Eq. (17), in turn, simplifies Eqs. (13) and (16), respectively, to

Dimensionless evolutionary power spectral density as a
function of dimensionless frequency for various

In this work, the spectrum of red noise is used to evaluate Eqs. (18) and (19)
explicitly. The analysis of discharge variability in this section
assumes an exponential form for the autocovariance function of the random
fluctuations in the peak inflow rate (Jin and Duffy, 1994; Kumar and Duffy, 2009), namely,

Dimensionless variance of discharge fluctuations as a function of dimensionless time for various dimensionless temporal correlation scales of inflow rate fluctuations.

Upon substituting Eq. (20b) into Eq. (18) and integrating it over the
frequency domain, one obtains the following expression for the variance of
flow discharge fluctuations as

Dimensionless variance of discharge fluctuations as a function of dimensionless distance from the upstream boundary.

Variation of flow discharge with the distance from the upstream boundary is depicted in Fig. 3 according to Eq. (21). As noted in the figure, the variability of flow discharge grows monotonically with distance, implying that due to the naturally inherent variability of lateral inflow, uncertainty in the flow discharge calculations from a deterministic model increases with the distance from the upstream boundary. In other words, the prediction of flow discharge distribution based on the deterministic simulation results is subject to the largest uncertainty in the downstream region. The downstream region is important in most real applications of modeling, and Eq. (21) provides a way of assessing the variation around the deterministic model prediction.

Many practical applications involving prediction over a large scale require measurement of uncertainty. Standard deviation is the best way to accomplish that. In this sense, the prediction results from a deterministic model are treated as the mean values. The mean value plus one standard deviation (square root of Eq. (21) provides a rational basis for extrapolating relatively small-scale field observations to these large space scales. Moreover, the likelihood of the flow discharge falling in the range of one standard deviation greater and smaller than the mean is about 68.27 %.

The problem of fluctuations in flow discharge in open channels in response to temporal changes in lateral inflow rate is investigated stochastically for a finite flow domain. In this study, the inflow perturbation field is modeled as a temporally stationary random process. For a complete stochastic description of flow discharge variability, expressions for the covariance function and evolutionary power spectral density of the random flow discharge perturbation process are developed. These expressions are obtained using a spectral representation theory. The variance relation developed here provides a rational basis for quantifying the uncertainty in applying the deterministic model.

This work represents an initial step in stochastic study of the effect of temporal variation of lateral inflow on the stream flow discharge variability. To take the advantage of a closed-form solution, the linearized diffusion wave equation (Eq. 2) is therefore used as the starting point for this research. It is important to recognize that the results developed in this work are valid only for the case of small variations in flow discharge around an initially uniform flow regime.

It is found from our closed-form expressions that the discharge variability in stream channels induced by the temporal changes in lateral inflow rate increases gradually with time toward its asymptotic value at large time. A larger temporal correlation scale of inflow rate fluctuations, which is of a more persistence of inflow perturbation process, will introduce more variability of the flow discharge. The increase of discharge variability with the distance from the upstream boundary suggests that prediction of flow discharge distribution in channels using a deterministic model is subject to large uncertainty at the downstream reach of the stream.

This research work is supported by the Taiwan Ministry of Science Technology under the grants NSC 101-2221-E-009-105-MY2, 102-2221-E-009-072-MY2 and NSC 102-2218-E-009-013-MY3. We are grateful to the editor Zhongbo Yu and anonymous referees for constructive comments that improved the quality of the work. Edited by: Z. Yu