A recent research focus of our group has been the development and use of synthetic streamflow generators. There are many tools one might use to generate synthetic streamflows and it may not be obvious which is right for a specific application or what the inherent limitations of each method are. More fundamentally, it may not be obvious why it is desirable to generate synthetic streamflows in the first place. This will be the first in a series of blog posts on the synthetic streamflow generators in which I hope to sketch out the various categories of generation methods and their appropriate use as I see it. In this first post we’ll focus on the motivation and history behind the development of synthetic streamflow generators and broadly categorize them.

**Why should we use synthetic hydrology?**

The most obvious reason to use synthetic hydrology is if there is little or no data for your system (see Lamontagne, 2015). Another obvious reason is if you are trying to evaluate the effect of hydrologic non-stationarity on your system (Herman et al. 2015; Borgomeo et al. 2015). In that case you could use synthetic methods to generate flows reflecting a shift in hydrologic regime. But are there other reasons to use synthetic hydrology?

In water resources systems analysis it is common practice to evaluate the efficacy of management or planning strategies by simulating system performance over the historical record, or over some critical period. In this approach, new strategies are evaluated by asking the question: *How well would we have done with your new strategy?*

This may be an appealing approach, especially if some event was particularly traumatic to your system. But is this a robust way of evaluating alternative strategies? It’s important to remember that any hydrologic record, no matter how long, is only a single realization of a stochastic process. Importantly, drought and flood events emerge as the result of specific sequences of events, unlikely to be repeated. Furthermore, there is a 50% chance that the worst flood or drought in an *N *year record will be exceeded in the next *N* years. Is it well advised to tailor our strategies to past circumstances that will likely never be repeated and will as likely as not be exceeded? As Lettenmaier et al. [1987] reminds us “Little is certain about the future except that it will be unlike the past.”

Even under stationarity and even with long hydrologic records, the use of synthetic streamflow can improve the efficacy of planning and management strategies by exposing them to larger and more diverse flood and drought than those in the record (Loucks et al. 1981; Vogel and Stedinger, 1988; Loucks et al. 2005). Figure 7.12 from Loucks et al. 2005 shows a typical experimental set-up using synthetic hydrology with a simulation model. Often our group will wrap an optimization model like Borg around this set up, where the system design/operating policy (bottom of the figure) are the decision variables, and the system performance (right of the figure) are the objective(s).

**What are the types of generators?**

Many synthetic streamflow generation techniques have been proposed since the early 1960s. It can be difficult for a researcher or practitioner to know which method is best suited to the problem at hand. Thus, we’ll start with a very broad characterization of what is out there, then proceed to some history.

Broadly speaking there are two approaches to generating synthetic hydrology: *indirect* and *direct*. The *indirect* approach generates streamflow by synthetically generating the forcings to a hydrologic model. For instance one might generate precipitation and temperature series and input them to a hydrologic model of a basin (e.g. Steinschneider et al. 2014). In contrast, *direct* methods use statistical techniques to generate streamflow timeseries directly.

The *direct* approach is generally easier to apply and more parsimonious because it does not require the selection, calibration, and validation of a separate hydrologic model (Najafi et al. 2011). On the other hand, the *indirect* approach may be desirable. Climate projections from GCMs often include temperature or precipitation changes, but may not describe hydrologic shifts at a resolution or precision that is useful. In other cases, profound regime shifts may be difficult to represent with statistical models and may require process-driven models, thus necessitating the *indirect* approach.

Julie’s earlier series focused on *indirect* approaches, so we’ll focus on the *direct* approach. Regardless of the approach many of the methods are same. In general generator methods can be divided into two categories: *parametric* and *non-parametric*. *Parametric* methods rely on a hypothesized statistical model of streamflow whose parameters are selected to achieve a desired result (Stedinger and Taylor, 1982a). In contrast *non-parametric* methods do not make strong structural assumptions about the processes generating the streamflow, but rather rely on re-sampling from the hydrologic record in some way (Lall, 1995). Some methods combine *parametric* and *non-parametric* techniques, which we’ll refer to as semi-parametric (Herman et al. 2015).

Both *parametric* and *non-parametric* methods have advantages and disadvantages. *Parametric* methods are often parsimonious, and often have analytical forms that allow easy parameter manipulation to reflect non-stationarity. However, there can be concern that the underlying statistical models may not reflect the hydrologic reality well (*Sharma et al*.** **1997). Furthermore, in multi-dimensional, multi-scale problems the proliferation of parameters can make parametric models intractable (Grygier and Stedinger, 1988). Extensive work has been done to confront both challenges, but they may lead a researcher to adopt a *non-parametric* method instead.

Because many *non-parametric *methods ‘re-sample’ flows from a record, realism is not generally a concern, and most re-sampling schemes are computationally straight forward (relatively speaking). On the other hand, manipulating synthetic flows to reflect non-stationarity may not be as straightforward as a simple parameter change, though methods have been suggested (Herman et al. 2015; Borgomeo et al. 2015). More fundamentally, because *non-parametric* methods rely so heavily on the data, they require sufficiently long records to ensure there is enough hydrologic variability to sample. Short records can be a concern for *parametric* methods as well, though parametric uncertainty can be explicitly considered in those methods (Stedinger and Taylor, 1982b). Of course, *parametric* methods also have structural uncertainty that non-parametric models largely avoid by not assuming an explicit statistical model.

In the coming posts we’ll dig into the nuances of the different methods in greater detail.

**A historical perspective**

The first use of synthetic flow generation seems to have been by Hazen [1914]. That work attempted to quantify the reliability of a water supply by aggregating the streamflow records of local streams into a 300-year ‘synthetic record.’ Of course the problem with this is that the cross-correlation between concurrent flows rendered the effective record length much less than the nominal 300 years.

Next Barnes [1954] generated 1,000 years of streamflow for a basin in Australia by drawing random flows from a normal distribution with mean and variance equal to the sample estimates from the observed record. That work was extended by researchers from the Harvard Water Program to account for autocorrelation of monthly flows (Maass et al., 1962; Thomas and Fiering, 1962). Later work also considered the use of non-normal distributions (Fiering, 1967), and the generation of correlated concurrent flows at multiple sites (Beard, 1965; Matalas, 1967).

Those early methods relied on first-order autoregressive models that regressed flows in the current period on the flows of the previous period (see Loucks et al.’s Figure 7.13 below). Box and Jenkins [1970] extended those methods to autoregressive models of arbitrary order, moving average models of arbitrary order, and autoregressive-moving average models of arbitrary order. Those models were the focus of extensive research over the course of the 1970s and 1980s and underpin many of the parametric generators that are widely used in hydrology today (see Salas et al. 1980; Grygier and Stedinger, 1990; Salas, 1993; Loucks et al. 2005).

By the mid-1990s, *non-parametric* methods began to gain popularity (Lall, 1995). While much of this work has its roots in earlier work from the 1970s and 1980s (Yakowitz, 1973, 1979, 1985; Schuster and Yakowitz, 1979; Yakowitz and Karlsson, 1987; Karlson and Yakowitz, 1987), improvements in computing and the availability of large data sets meant that by the mid-1990s non-parametric methods were feasible (Lall and Sharma, 1996). Early examples of *non-parametric *methods include block bootstrapping (Vogel and Shallcross, 1996), k-nearest neighbor (Lall and Sharma, 1996), and kernel density methods (Sharma et al.** **1997). Since that time extensive research has made improvement to these methods, often by incorporating parametric elements. For instance, Srinivas and Srinivasan (2001, 2005, and 2006) develop a hybrid autoregressive-block bootstrapping method designed to improve the bias in lagged correlation and to generate flows other than the historical, for multiple sites and multiple seasons. K-nearest neighbor methods have also been the focus of extensive research (Rajagopalan and Lall, 1999; Harrold et al. 2003; Yates et al. 2003; Sharif and Burn, 2007; Mehortra and Sharma, 2006; Prairie et al. 2006; Lee et al. 2010, Salas and Lee, 2010, Nowak et al., 2010), including recent work by our group (Giuliani et al. 2014).

Emerging work focuses on stochastic streamflow generation using copulas [*Lee and Salas*, 2011; Fan et al. 2016], entropy theory bootstrapping [*Srivastav and Simonovic*, 2014], and wavelets [*Kwon et al*. 2007; Erkyihun et al., 2016], among other methods.

In the following posts I’ll address different challenges in stochastic generation [e.g. long-term persistence, parametric uncertainty, multi-site generation, seasonality, etc.] and the relative strengths and shortcomings of the various methods for addressing them.

**Works Cited**

Barnes, F. B., Storage required for a city water supply, J. Inst. Eng. Australia 26(9), 198-203, 1954.

Beard, L. R., Use of interrelated records to simulate streamflow, J. Hydrol. Div., ASCE 91(HY5), 13-22, 1965.

Borgomeo, E., Farmer, C. L., and Hall, J. W. (2015). “Numerical rivers: A synthetic streamflow generator for water resources vulnerability assessments.” Water Resour. Res., 51(7), 5382–5405.

Y.R. Fan, W.W. Huang, G.H. Huang, Y.P. Li, K. Huang, Z. Li, Hydrologic risk analysis in the Yangtze River basin through coupling Gaussian mixtures into copulas, Advances in Water Resources, Volume 88, February 2016, Pages 170-185.

Fiering, M.B, Streamflow Synthesis, Harvard University Press, Cambridge, Mass., 1967.

Giuliani, M., J. D. Herman, A. Castelletti, and P. Reed (2014), Many-objective reservoir policy identification and refinement to reduce policy inertia and myopia in water management, Water Resour. Res., 50, 3355–3377, doi:10.1002/2013WR014700.

Grygier, J.C., and J.R. Stedinger, Condensed Disaggregation Procedures and Conservation Corrections for Stochastic Hydrology, Water Resour. Res. 24(10), 1574-1584, 1988.

Grygier, J.C., and J.R. Stedinger, SPIGOT Technical Description, Version 2.6, 1990.

Harrold, T. I., Sharma, A., and Sheather, S. J. (2003). “A nonparametric model for stochastic generation of daily rainfall amounts.” Water Resour. Res., 39(12), 1343.

Hazen, A., Storage to be provided in impounding reservoirs for municipal water systems, Trans. Am. Soc. Civ. Eng. 77, 1539, 1914.

Herman, J.D., H.B. Zeff, J.R. Lamontagne, P.M. Reed, and G. Characklis (2016), Synthetic Drought Scenario Generation to Support Bottom-Up Water Supply Vulnerability Assessments, Journal of Water Resources Planning & Management, doi: 10.1061/(ASCE)WR.1943-5452.0000701.

Karlsson, M., and S. Yakowitz, Nearest-Neighbor methods for nonparametric rainfall-runoff forecasting, Water Resour. Res., 23, 1300-1308, 1987.

Kwon, H.-H., U. Lall, and A. F. Khalil (2007), Stochastic simulation model for nonstationary time series using an autoregressive wavelet decomposition: Applications to rainfall and temperature, Water Resour. Res., 43, W05407, doi:10.1029/2006WR005258.

Lall, U., Recent advances in nonparametric function estimation: Hydraulic applications, U.S. Natl. Rep. Int. Union Geod. Geophys. 1991- 1994, Rev. Geophys., 33, 1093, 1995.

Lall, U., and A. Sharma (1996), A nearest neighbor bootstrap for resampling hydrologic time series, Water Resour. Res. 32(3), pp. 679-693.

Lamontagne, J.R. 2015,Representation of Uncertainty and Corridor DP for Hydropower 272 Optimization, PhD edn, Cornell University, Ithaca, NY.

Lee, T., J. D. Salas, and J. Prairie (2010), An enhanced nonparametric streamflow disaggregation model with genetic algorithm, Water Resour. Res., 46, W08545, doi:10.1029/2009WR007761.

Lee, T., and J. Salas (2011), Copula-based stochastic simulation of hydrological data applied to Nile River flows, Hydrol. Res., 42(4), 318–330.

Lettenmaier, D. P., K. M. Latham, R. N. Palmer, J. R. Lund and S. J. Burges, Strategies for coping with drought Part II: Planning techniques for planning and reliability assessment, EPRI P-5201, Final Report Project 2194-1, June 1987.

Loucks, D.P., Stedinger, J.R. & Haith, D.A. 1981, Water Resources Systems Planning and Analysis, 1st edn, Prentice-Hall, Englewood Cliffs, N.J.

Loucks, D.P. et al. 2005, Water Resources Systems Planning and Management: An Introduction to Methods, Models and Applications, UNESCO, Delft, The Netherlands.

Maass, A., M. M. Hufschmidt, R. Dorfman, H. A. Thomas, Jr., S. A. Marglin and G. M. Fair,

Design of Water Resource Systems, Harvard University Press, Cambridge, Mass., 1962.

Matalas, N. C., Mathematical assessment of synthetic hydrology, Water Resour. Res. 3(4), 937-945, 1967.

Najafi, M. R., Moradkhani, H., and Jung, I. W. (2011). “Assessing the uncertainties of hydrologic model selection in climate change impact studies.” Hydrol. Process., 25(18), 2814–2826.

Nowak, K., J. Prairie, B. Rajagopalan, and U. Lall (2010), A nonparametric stochastic approach for multisite disaggregation of annual to daily

streamﬂow, Water Resour. Res., 46, W08529, doi:10.1029/2009WR008530.

Nowak, K., J. Prairie, B. Rajagopalan, and U. Lall (2010), A nonparametric stochastic approach for multisite disaggregation of annual to daily

streamﬂow, Water Resour. Res., 46, W08529, doi:10.1029/2009WR008530.

Nowak, K., J. Prairie, B. Rajagopalan, and U. Lall (2010), A nonparametric stochastic approach for multisite disaggregation of annual to daily

streamﬂow, Water Resour. Res., 46, W08529, doi:10.1029/2009WR008530.

Nowak, K., J. Prairie, B. Rajagopalan, and U. Lall (2010), A nonparametric stochastic approach for multisite disaggregation of annual to daily streamflow, Water Resour. Res., 46, W08529, doi:10.1029/2009WR008530.

Prairie, J. R., Rajagopalan, B., Fulp, T. J., and Zagona, E. A. (2006). “Modified K-NN model for stochastic streamflow simulation.” J. Hydrol. Eng., 11(4), 371–378.

Rajagopalan, B., and Lall, U. (1999). “A k-nearest-neighbor simulator for daily precipitation and other weather variables.” Water Resour. Res., 35(10), 3089–3101.

Salas, J. D., J. W. Deller, V. Yevjevich and W. L. Lane, Applied Modeling of Hydrologic Time Series, Water Resources Publications, Littleton, Colo., 1980.

*Salas*, J.D., 1993, Analysis and Modeling of *Hydrologic* Time Series, *Chapter 19* (72 p.) in The McGraw Hill *Handbook of Hydrology*, D.R. Maidment, Editor.

Salas, J.D., T. Lee. (2010). Nonparametric Simulation of Single-Site Seasonal Streamflow, J. Hydrol. Eng., 15(4), 284-296.

Schuster, E., and S. Yakowitz, Contributions to the theory of nonparametric regression, with application to system identification, Ann. Stat., 7, 139-149, 1979.

Sharif, M., and Burn, D. H. (2007). “Improved K-nearest neighbor weather generating model.” J. Hydrol. Eng., 12(1), 42–51.

Sharma, A., Tarboton, D. G., and Lall, U., 1997. “Streamflow simulation: A nonparametric approach.” Water Resour. Res., 33(2), 291–308.

Srinivas, V. V., and Srinivasan, K. (2001). “A hybrid stochastic model for multiseason streamflow simulation.” Water Resour. Res., 37(10), 2537–2549.

Srinivas, V. V., and Srinivasan, K. (2005). “Hybrid moving block bootstrap for stochastic simulation of multi-site multi-season streamflows.” J. Hydrol., 302(1–4), 307–330.

Srinivas, V. V., and Srinivasan, K. (2006). “Hybrid matched-block bootstrap for stochastic simulation of multiseason streamflows.” J. Hydrol., 329(1–2), 1–15.

Roshan K. Srivastav, Slobodan P. Simonovic, An analytical procedure for multi-site, multi-season streamflow generation using maximum entropy bootstrapping, Environmental Modelling & Software, Volume 59, September 2014a, Pages 59-75.

Stedinger, J. R. and M. R. Taylor, Sythetic streamflow generation, Part 1. Model verification and validation, Water Resour. Res. 18(4), 909-918, 1982a.

Stedinger, J. R. and M. R. Taylor, Sythetic streamflow generation, Part 2. Parameter uncertainty,Water Resour. Res. 18(4), 919-924, 1982b.

Steinschneider, S., Wi, S., and Brown, C. (2014). “The integrated effects of climate and hydrologic uncertainty on future flood risk assessments.” Hydrol. Process., 29(12), 2823–2839.

Thomas, H. A. and M. B. Fiering, Mathematical synthesis of streamflow sequences for the analysis of river basins by simulation, in Design of Water Resource Systems, by A. Maass, M. Hufschmidt, R. Dorfman, H. A. Thomas, Jr., S. A. Marglin and G. M. Fair, Harvard University Press, Cambridge, Mass., 1962.

Vogel, R.M., and J.R. Stedinger, The value of stochastic streamflow models in over-year reservoir design applications, Water Resour. Res. 24(9), 1483-90, 1988.

Vogel, R. M., and A. L. Shallcross (1996), The moving block bootstrap versus parametric time series models, Water Resour. Res., 32(6), 1875–1882.

Yakowitz, S., A stochastic model for daily river flows in an arid region, Water Resour. Res., 9, 1271-1285, 1973.

Yakowitz, S., Nonparametric estimation of markov transition functions, Ann. Stat., 7, 671-679, 1979.

Yakowitz, S. J., Nonparametric density estimation, prediction, and regression for markov sequences J. Am. Stat. Assoc., 80, 215-221, 1985.

Yakowitz, S., and M. Karlsson, Nearest-neighbor methods with application to rainfall/runoff prediction, in Stochastic Hydrology, edited by J. B. Macneil and G. J. Humphries, pp. 149-160, D. Reidel, Norwell, Mass., 1987.

Yates, D., Gangopadhyay, S., Rajagopalan, B., and Strzepek, K. (2003). “A technique for generating regional climate scenarios using a nearest-neighbor algorithm.” Water Resour. Res., 39(7), 1199.

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