WATER RESOURCES RESEARCH, VOL. 28,NO. 7, PAGES 1823-1831, JULY 1992 Demodulation-Remodulation Revisited: Theoryand Application JOHNW. HUMPHRIES AND PEDRO J. RESTREPO Center for Advanced Decision Support for Water and Environmental Systems, University of Colorado, Boulder JUAN B. VALDI•S Department of Civil Engineering and Climate System Research Program, Texas A&M University, College Station In modeling periodic hydrologic series, theamplitudes andphase angles of the harmonics reflect the magnitudes and timing of the seasonal peaks. Demodulation-remodulation capturesthe temporal variations in these stochastic parameters, thereby providinguseful informationand more accurate models than traditional Fourier seriesmodels.Unfortunately,this technique has been neglected by hydrologicmodelers.In this paper, application of the technique to a synthetic seriesvalidates the theory. The technique is then appliedto the streamflow recordof the Uda Walawe River which exhibits a strong bimodalannual pattern. Fourteen models are developed, providing a comparison betweenmovingaverage filters commonly used in demodulation studies, the Fourier filter usedin Fourier series analysis, and composite modelswhich incorporate both filters. In addition, the development of a prototype forecasting filter demonstrates the viability of using demodulation- remodulation for forecasting. INTRODUCTION Due to seasonal climatic variability, many hydrologic proc•esses are characterized by an annual periodicity. In regions subject to heavy snowfall, such as the mountain stal[es of the western United States, the streamflow patterns reflect the annual spring snowmelt. Similarly, many tropical regions aresubject to seasonal monsoons which cause fairly predictable fluctuations in rainfall. Fourier series analysis, one of the traditional methods used to investigate such periodic series, seeks to describe the series mathematically as a summationof sine and cosine functions representing periodic components of various frequencies. One of the underlying assumptions in Fourier series analysis is that the amplitudes and phase angles ofthese components, known as harmonics, are constant. Whendiscussing hydrologic series suchas streamflow, however, the amplitudes and phase angles of theharmonics reflect the magnitude and timing of the seasonal peaks resulting from snowmelt or monsoon rains.Knowledge of the variable strength and temporal shifts of these peaks could be very beneficial in managing water resources and may provide more accurate models of hydrologic series. Demodulation provides a means for investigating the variations in amplitude and phase of theharmonics. These parameters can then be remodulated to provide a more accurate representation of the original series than thatob- tained from the constant parameters usedwith Fourier series. The demodulation-remodulation process may also provide a means for predicting the futurevalues of the parameters and thereby allow more accurate forecasts ofthe hydrologic process. The demodulation technique was first suggested by Tukey [1961]. Granger and Hatanaka [1964] provide a more thor- Copyright 1992 by theAmerican Geophysical Union. Paper number92WR00594. 0043-! 397/92/92WR-00594505.00 ough discussion which incorporates remodu!ation for simu- lationexperiments with economic time series. Rodriguez- Iturbe et al. [1971] employed demodulation in investigating the characteristics of the annual harmonic in monthly streamflowrecords from three sites in the western United States.The demodulation resultswere then remodulatedand used in simulating the streamflow at one of the sites. Anderson [1975]and Gonzalez-Sanabria et al. [1979] are more recent examples of theuseof demodulation in analyz- ing hydrologic time series. Because demodulation-remodulation is so well suited to modeling periodic streamflow series, it seems odd that it has received so little attention. Onepossible explanation for this neglect is that the only published study which has applied the technique tosimulation achieved rather discouraging results. In thestudy byRodriguez-Iturbe et aI. [1971], theresearch- ers investigated four different models andfound that the model incorporating demodulation-remodu!ation did not per- form as well as models using the traditional Fourier series approach. Demodu!ation-remodulation is generally pre- sented as anadvanced application of Fourier series analysis [Gabel and Roberts, 1987], and thepoor results of the1971 study failed to justify this increased complexity. Further- more, thetheoretical foundations of thetechnique have not been presented in the context of hydrologic series to make the procedure more accessible to hydrologists. With this paper, the authors seek to address these issues and revive an interest in demodulation-remodu!ation within thecommunity of hydrologic modelers. Weprovide a thor- ough explanation of thetheory in relation to hydrologic modeling, we explain that Fourier series analysis isactually a special case ofthe more general demodulation-remodula- tion technique, and we present positive results from simula- tion studies involving synthetic and actual streamflow series. In addition, we present a new type of filter which demon- strates the viability of forecasting with demodulation- remodulation. 1823