Stochastically generated streamflow time series are used in various applications of water resource planning and management. These applications include water and reservoir management, the determination of the dimensions of hydraulic structures such as reservoirs, and the estimation of hydrological extremes such as droughts and floods. Stochastically generated time series mimic the characteristics of observed data and represent sets of plausible realizations of streamflow sequences . They are essential for many uncertainty studies in hydrology because they can serve as input for deterministic water system models in which they allow for the propagation of natural variability and uncertainty .

Stochastic models for the generation of synthetic streamflow time series need to fulfil certain requirements. They should reproduce both the marginal distribution of observed streamflow time series as well as their temporal dependence structure . Temporal dependence encompasses both short- and long-range dependence. While short-range dependence typically refers to the dependence of daily streamflow values measured within a few days, long-range dependence refers to dependencies across months or years. This temporal dependence has been found to depend on magnitude, in that low values have stronger dependence than high values . A proper representation of this long-range dependence is of particular importance in studies where storage in reservoirs is of interest . If one is interested in extreme events, the model should allow for the generation of values that go beyond the magnitude of those observed . This requires the choice of a suitable theoretical marginal distribution. Streamflow typically exhibits a skewed distribution, which requires the use of a three- or more-than-three-parameter distribution . Studies looking at individual hydrological events such as floods or droughts require a daily resolution. Therefore, the stochastic model should allow for outputs at such a fine temporal resolution. Often, study regions encompass several sites whose streamflows are correlated. Consequently, the model ideally not only allows for the simulation of streamflow at individual sites, but also for the joint simulation of streamflow at multiple sites, taking into account their spatio-temporal dependence. From a practitioner's point of view, the model should not only reproduce the characteristics of the observed data, but it should also be simple .

Many different approaches have been proposed for the stochastic simulation of streamflow time series, each able to fulfil some but usually not all of the desired properties listed above. One commonly used approach is the use of a synthetic weather generator in combination with a rainfall–runoff model . This approach is affected by uncertainties due to hydrological model selection and calibration, which can be avoided by using direct synthetic streamflow generation approaches . According to , the development of such direct approaches consists of the following steps: (1) selection of a model for seasonal marginal distributions, (2) selection of a model for spatial and temporal dependence, and (3) validation of the model. Different groups of direct approaches exist which are distinct in terms of their flexibility regarding marginal distributions and temporal dependence structures.

A first group of models consists of parametric models such as autoregressive moving average (ARMA) models and their modifications. While these models are commonly used in stochastic hydrology, they only allow for modelling of short-range dependence because their autocorrelation decreases strongly with increasing lag time . This means they guarantee neither the reproduction of observed persistence of annual flows nor the correlation structure among flows in different months . This makes them unsuitable for applications where long-range dependence is important . However, AR models can be used to generate seemingly long-memory processes if a parametric autocorrelation structure is used to fit the data . A second group of parametric models is based on the temporal disaggregation of annual series and enables the representation of long-range dependence . These models include fractional Gaussian noise models , fast fractional Gaussian noise models , broken line models , and fractional autoregressive integrated moving average models . Disaggregation models can be extended to multi-site applications . However, this group of models has been shown to exhibit parameter estimation problems and only allows for the representation of a narrow range of autocorrelation functions . A third group of models is nonparametric in its approach and includes kernel density estimation and various bootstrap approaches. The latter include simple bootstrap, which is only useful if data are uncorrelated, moving block-bootstrap, nearest-neighbour bootstrap , matched-block bootstrap , and maximum-entropy bootstrap , which also take lagged correlations into account. These nonparametric techniques resample from the data with perturbations and directly reproduce the characteristics of the original data . However, the reproduction of long-range dependence is difficult, and variance can be underestimated or overestimated . To allow for values that go beyond the observed distribution, proposed a model employing k-nearest-neighbour resampling with a gamma kernel perturbation. A further group of models consists of models that employ Markov chains and their variations. These models account for transition probabilities between different hydrological states and can be combined with nonparametric approaches such as k-nearest neighbours . They can be extended to multiple sites by scaling the simulated values at individual sites with spatially correlated random numbers .

Several alternatives to these well-established simulation procedures have been proposed, which allow for a flexible choice of marginal distributions. These include models where the temporal dependence structure is modelled with copula functions, which are, however, difficult to apply for higher orders of autocorrelation . Examples of new simulation procedures based on the Autoregressive to Anything (ARTA) model proposed by following are the SMARTA model by or the SPARTA model by , which employ Nataf's joint distribution model for the simulation of stochastic time series, representing both short- and long-range dependence. In addition, simulation schemes based on wavelet decomposition, which avoid assumptions about the temporal dependence structure, have been proposed by , , and . have shown how simulated annealing can be used to generate synthetic streamflow time sequences that represent possible climate-induced changes in user-specified streamflow properties.

All these previously mentioned models are based on the time domain. An alternative to time-domain models is frequency-domain models , which allow for the simulation of surrogate data with the same Fourier spectra as the raw data . Such methods are based on the randomization of the phases of the Fourier transformation and have been commonly applied in hypothesis testing, when identifying nonlinearity in time series , and in trend detection . We hereafter refer to such methods, which are also known as amplitude-adjusted Fourier transformations (AAFTs) , as phase randomization simulations. used an iterative AAFT method to generate binary series of rainfall occurrence and non-occurrence. An extension of the amplitude-adjusted Fourier transformation has been presented by , who applied randomization procedures to wavelet-decomposed signals to generate surrogate data. In hydrology, phase randomization simulation has rarely been applied for purposes other than hypothesis testing even though it has desirable properties which make it suitable for a wider range of applications. Indeed, its implementation is relatively simple, it can simulate time series with both short- and long-range dependence, and it can be extended to multiple sites. However, its application is often limited to the reproduction of the empirical distribution of the data. We here propose the use of phase randomization simulation for the stochastic generation of streamflow time series at individual and multiple sites. To allow for non-empirical distributions, we combine the data simulated by phase randomization with the flexible, four-parameter kappa distribution introduced by as a generalization of the three-parameter kappa distribution suggested by . The stochastic streamflow generation approach shall represent a flexible tool which is easy to apply and generalizable to different contexts. This is enabled by combining a nonparametric time dependence model with a flexible four-parameter distribution. The simulation approach can be tailored to the specific problem at hand and be used for various water resource management applications.

We now turn to some theoretical background on Fourier transformation and phase randomization. For a more detailed introduction to the Fourier transformation, the reader is referred to textbooks by or . We then discuss the use of phase randomization for the stochastic generation of streamflow time series. For illustration purposes, we apply and validate the approach on a set of four catchments in Switzerland. Finally, we discuss potential applications of the simulation approach.

The basic idea behind all surrogate methods is to randomize the Fourier phases of the underlying (hydrological) process. The Fourier transformation converts a time-domain signal into a frequency-domain signal, which is complex-valued. This transformation may be depicted as a decomposition of the time series into sine and cosine waves of different amplitude, phase, and period . In the frequency domain, the power spectral density (power spectrum) expresses the same information in cycles as the autocovariance function expresses in lags in the time domain. The periodogram, the empirical counterpart of the power spectrum, shows high values at those frequencies which correspond to strong periodic components .

The surrogate approach utilizes the property that realizations of linear Gaussian processes differ only in their Fourier phases and not their power spectrum. It preserves the autocorrelation structure of the raw series by conserving its power spectrum through phase randomization. The procedure consists of three main steps . In the first step, the streamflow series is converted from the time domain to the frequency domain by Fourier transformation . In this frequency domain, the data are represented by the phase angle and the power spectrum, as represented by the periodogram. The phase angle θ of the power spectrum is uniformly distributed over the range of -π to π. In the second step, the phases in the phase spectrum are randomized while the power spectrum is preserved. In the third step, the inverse Fourier transformation is applied to transform the data from the frequency domain back to the temporal domain .

The Fourier transformation of a given time series x=(x1, …, xt, …, xn) of length n is 1f(ω)=12πn∑t=1ne-iωtxt,-π≤ω≤π, where i=-1 is the imaginary unit. The original time series can be recovered by the back transformation 2xt=2πn∑j=1neiωjtfωj,t=1,2,…,n, if the transformation is calculated for discrete frequencies ωj=2π/n, j=1, 2, …, n. The Fourier transformation surrogate method constructs a new time series yt with the same periodogram as the observations. Apart from this, the new series are statistically independent of xt. This can be achieved by fixing the Fourier amplitudes |f(ωj)| and replacing the Fourier phases ϕ(ωj)=arg(f(ωj)) with uniformly distributed random numbers ϕrand(ωj)∈[-π, π]. A new realization is given by 3yt=2πn∑j=1neiωjtfωjeiϕrandωj. The surrogate data consist of the same values as the original data in another temporal order but with the same time dependence structure as the original data . The approach can be extended to multiple sites by multiplying the phases of each site by the same set of random phases. This is possible because the cross-spectrum, which describes the cross-correlation of the data in the frequency domain, reflects relative phases only .

The stochastic streamflow generator based on phase randomization has been shown to produce realistic streamflow time series with respect to both distributional properties and temporal correlation. Compared to models commonly used for the stochastic generation of streamflow time series, such as autoregressive moving average models, the simulation approach presented here reproduces not only short-range, but also long-range dependence. However, the representation of this dependence is limited to ranges within the length of the observed time series. Instead of producing one long time series, the simulation procedure allows for the simulation of multiple series of the same length as the original series. The use of ensembles of the same length as the observed time series might not be equivalent to using a long time series. Still, long-range dependence features may not be generated in either case since the model is fitted based on a limited number of years of observations. While the reproduction of the temporal dependence was well reproduced here, this is not necessarily the case under all conditions. have shown that any nonlinear transformation of a Gaussian time series, which is done during back transformation, reduces the strength of the linear correlations in the time series as expressed by Pearson's correlation coefficient and preserves only rank correlations. If one is working with heavy-tailed and zero inflated marginals (as present when looking at intermittent processes), it can happen that autocorrelations are reduced during back transformation .

Phase randomization was here combined with the flexible four-parameter kappa distribution, which was found to effectively represent daily streamflow values. The distribution of daily flows was found to be modelled well in all seasons. However, the use of one distribution per day has the disadvantage of introducing a lot of parameters, which makes the model non-parsimonious . If the user is not reliant on the generation of unobserved values, he/she might use the empirical instead of theoretical kappa distribution for back transformation instead. The use of the kappa distribution allows us to generate values that go beyond the range of observed values, which would not be the case if the empirical distribution was used. This ability of the generator to extrapolate extremes makes it suitable for applications where extreme events such as floods and droughts are of interest.

The generator can, on the one hand, be used to simulate streamflow at individual sites, and, on the other, to simulate jointly at multiple sites, which is not necessarily the case for other existing models. Its application to the example catchments, however, resulted in somewhat underestimated cross-correlations between stations. This underestimation can be explained by the fact that phase randomization preserves the cross-correlation in the normal domain but not necessarily in the domain of the original distribution. This cannot be overcome even if the simulation run which best reproduces these cross-correlations is extracted from a large set of simulations. However, showed that estimators of the autocorrelation and cross-correlation of flows which do not match the historical sample estimates often provide more accurate estimates of the true but unknown correlations. Still, there are several potential avenues for improving the representation of cross-correlation. A first possibility would be the use of phase annealing . Phase annealing modifies the Fourier phases in an iterative way in order to optimize certain statistics, such as the cross-correlation function, and makes it possible to take covariates into consideration for the generation of time series. However, using phase annealing increases the computational effort. A second possibility was presented by , who only randomized the phases corresponding to the wavelet coefficients lying above a certain threshold. He suggested fixing the large wavelet scales if one wanted to ensure that the low-frequency behaviour between the observations and simulated series remains the same. This can indeed be a solution for retaining the cross-correlation between two series. However, it comes with the disadvantage that the temporal structure of the simulated series is not very variable from the one of the observed series anymore. A third possibility is the introduction of functions correcting for the phase differences between two series as done by , who applied this approach to correct for biases across multiple atmospheric variables derived from global circulation models. Another possibility for addressing the underestimation of cross-correlation would be the inflation of the cross-spectrum in the original domain in order to allow for a certain target cross-correlation after back transformation. To do so, transformation approaches have been introduced, which inflate the original process, which should after the back transformation to the original domain result in a process with a target distribution and correlation structure . An additional disadvantage of the method presented here (and of most other approaches presented in the literature) is that time irreversibility, which has been shown to be significant at a daily scale , is not explicitly modelled.

The streamflow generator was here used on observed streamflow time series. The input time series, however, do not necessarily need to consist of observed values. One could also use the generator on streamflow simulated with a hydrological model. This extends its application to climate impact studies where a hydrological model is driven by meteorological time series generated with global and/or regional climate models. Alternatively, the representation of non-stationary conditions in the properties of the marginal distribution or the temporal dependence structure could also be achieved by adjusting the parameters of the marginal distribution or the frequency spectrum, respectively. Phase randomization simulation can potentially accommodate not only changing climate conditions, but also changes in land use or water extractions. The approach is not limited to the simulation of streamflow time series, but extends to other hydro-meteorological variables such as precipitation, evapotranspiration, or snowmelt. This would require the test and identification of a suitable marginal distribution. In the case of intermittent processes, mixed-type marginal distributions would need to be used . Distributions other than the kappa distribution can be used in PRSim by specifying a suitable (mixture) distribution. Spatio-temporal modelling of precipitation fields, for example, may be performed using a technique based on phase randomization. However, it must be noted that due to the large number of zero observations (specifically with fine temporal resolution), the normal score transformation can become non-unique. In this case, additional efforts are needed to preserve the spatial structure of precipitation.

The stochastic streamflow generator presented here represents a flexible tool for streamflow simulation at individual or multiple sites. It can be used for various applications such as the design of hydropower reservoirs, the assessment of flood risk, or the assessment of drought persistence and the estimation of the risk of multi-year droughts.