Analytical results for operational flash flood guidance

ati . G luff D 20 A warnings (e.g. Mogil et al., 1978). The basic idea is to Journal of Hydrology 31 E-mail address: [email protected]. Abstract The theoretical basis of developing operational flash flood guidance systems is studied using analytical methods. The Sacramento soil moisture accounting model is used operationally in the United States to produce flash flood guidance estimates of a given duration from threshold runoff estimates. Analytical results to (a) shed light on the properties of this model’s short- term surface runoff predictions under substantial rainfall forcing and (b) to facilitate flash flood computations in real time. The results derived pertain to the computation of pervious-area surface runoff from the upper zone tension and free water elements of the model, and to surface runoff from both the permanently impervious and the variable additional impervious area of the model that provides spatially distributed characteristics in the production of surface runoff. The time to the production of surface runoff is also computed on the basis of the model parameters and initial water element conditions and is related to the flash flood guidance duration. Analytical solutions for time to saturation and surface runoff are also derived for the case of saturation-excess models, and these results are compared to those derived for the Sacramento model. Various characteristics of the flash flood guidance to threshold runoff relationship are discussed and considerations for real-time application are offered. Uncertainty analysis of the threshold runoff to flash flood guidance transformation is also performed. q 2005 Elsevier B.V. All rights reserved. Keywords: Flash floods; Hydrologic modeling; Real-time prediction; Model uncertainty 1. Introduction The present paper addresses theoretical under- pinnings of operational flash flood guidance compu- tations. Since the 1970s the US National Weather Service relies routinely on flash flood guidance computations to produce flash flood watches and and predicted (for forecasting) rainfall volume of a given duration and over a given catchment to a characteristic volume of rainfall for that duration and catchment that generates bankfull flow conditions at the catchment outlet. If the nowcast or forecast rainfall volume is greater than the characteristic rainfall volume then flooding in the catchment is Analytical results for oper Konstantine P Hydrologic Research Center, 12780 High B Received 2 March 2004; revised onal flash flood guidance eorgakakos* rive, Suite 250, San Diego, CA 92130, USA pril 2005; accepted 4 May 2005 7 (2006) 81–103 www.elsevier.com/locate/jhydrol of the forecasts or nowcasts may be derived based on Bayes theorem (e.g. see Georgakakos, 1992 for a nowcasting example). 0022-1694/$ - see front matter q 2005 Elsevier B.V. All rights reserved. doi:10.1016/j.jhydrol.2005.05.009 * Tel.: C1 858 794 2726; fax: C1 858 792 2519. compare in real time observed (for nowcasting) likely. Formal probabilistic methods for deciding when to issue a warning on the basis of the uncertainty of Hy Nomenclature A catchment area b baseflow rate K.P. Georgakakos / Journal82 Determination of flash flood guidance in an operational environment in the United States requires the development of (a) estimates of threshold runoff volume of various durations, and (b) a soil moisture accounting model to develop the curves that relate CV coefficient of variation (standard deviation divided by mean) e evapotranspiration rate eF evapotranspiration rate from upper zone free water storage ep evapotranspiration demand rate eT evapotranspiration rate from upper soil zone tension water storage i precipitation rate pf percolation rate to lower zone storages p0 percolation rate constant parameter Ra flash flood guidance rf excess flow rate from upper zone tension water to upper zone free water storage Rs threshold runoff s Surface runoff saf surface runoff rate from additional imper- vious area sf surface runoff rate to channel network t time variable td duration of flash flood guidance tD time of rainfall duration tF time of saturation of upper zone free water storage ti forecast preparation time tI time to initiate surface runoff from initial soil water deficit conditions t0 initial time of ‘what-if’ integrations for flash flood guidance tq time of depletion of additional impervious area water tr duration of effective rainfall ts saturation time for saturation-excess soil moisture model tT time of saturation of upper zone tension water storage uf interflow rate from upper zone free water storage drology 317 (2006) 81–103 threshold runoff to flash flood guidance of a given duration as a function of soil moisture deficit (Sweeney et al. 1992). For the present analysis, threshold runoff, Rs, is the volume of effective rainfall of given duration, tr, that is generated over a given catchment of area A xA additional impervious area water storage in excess of upper zone tension water capacity xAF value of additional impervious area water storage in excess of upper zone tension water capacity for fully saturated upper zone soils xA0 initial value of additional impervious area water storage in excess of upper zone tension water capacity xF upper zone free water volume per unit area of catchment xF0 initial upper zone free water volume per unit area of catchment x0F upper zone free water capacity x0L lower zone tension water capacity x0T upper zone tension water capacity xT upper zone tension water volume per unit area of catchment xT0 initial upper zone tension water volume per unit area of catchment Z soil depth in saturation-excess soil moist- ure model a rate of interflow production (in inverse time units) b time variable area fraction of variable impervious area within the catchment b1 maximum area fraction of variable impervious area within the catchment b2 area fraction of permanently impervious area within the catchment q linearization parameter in saturation- excess soil moisture model baseflow qs soil moisture (vol/vol) 3 saturation soil moisture (vol/vol) and that is just enough to cause bankfull flow at the outlet of the draining stream. It is noted at the outset that threshold runoff, based on the concept of effective rainfall, also represents surface runoff volume with the characteristic that is just enough to realize bankfull conditions at the watershed outlet. Flash flood guidance, Ra, of duration td is defined as the volume of actual rainfall that generates the said threshold runoff. Flash flood guidance may be compared to observed or forecast rainfall volume of the same duration td for the formulation and issuance of warnings and watches in an operational environment. In general tdstr, with tr often being less than td because significant surface runoff generation may only occur after a period of initial wetting. In real time operations it is assumed that tdZtr. Earlier studies have developed national estimates of threshold runoff from digital spatial data and stream a continuous time form is in Georgakakos (1986). Koren et al. (2000) and Duan et al. (2001) link the parameters of the SAC model to watershed soil and land-cover characteristics. The good performance of the SAC model to reproduce high flow conditions under spatially lumped and spatially distributed implementations is most recently demonstrated by the results of the Distributed Modeling Intercompar- ison Project (DMIP) organized by the US National Weather Service Office of Hydrologic Development (Smith et al., 2004; Reed et al., 2004). As such it represents an example of a widely applicable model for the purposes of this work. It is noted that in addition to the operational flash flood guidance system of the US National Weather Service, very recently a second operational flash flood guidance system, this one regional in character, has been blem. n td an om th K.P. Georgakakos / Journal of Hydrology 317 (2006) 81–103 83 surveys on the basis of unit hydrograph theory (synthetic and geomorphologic) and steady uniform flow theory or flow estimates of a given return period (Carpenter et al., 1999). The present study produces and discusses analytical results pertaining to the production of surface runoff from the operational Sacramento soil moisture accounting model (here- tofore called SAC), used in the United States to convert threshold runoff estimates to flash flood guidance estimates. The discrete form of the model has been described in Burnash et al. (1973), while Fig. 1. Actual and ‘what if’ time axes for the flash flood guidance pro determine the relationship between a given rainfall volume of duratio axis starting from initial conditions at time t0 on that axis obtained fr actual time axis. reported for the seven countries of Central America (e.g. Georgakakos, 2004). In this case, a saturation-excess physically-based soil moisture accounting model is used to compute depth-integrated soil moisture deficit in real time. For generality and when appropriate in this paper, we specialize the Sacramento model results for such saturation-excess soil models. In the United States, the SAC model runs operationally over basins of area O (1000 km2) to produce streamflow estimates and forecasts at each of several forecast preparation times ti indicated in Fig. 1 Operational runs are performed on the actual time axis while runs to d the resultant surface runoff volume are performed on the ‘what if’ e operational runs of the model for the corresponding time ti on the on the ‘actual’ time axis. At the completion of the operational forecast run, the current estimates of the volumes in each of the model soil moisture compartments (upper zone tension water, upper zone free water, lower zone tension water, lower zone free primary and supplementary water, and additional impervious area water), valid at the forecast prep- aration time, are stored. To support flash flood computations and using these initial conditions, the model runs off-line in ‘what if’ scenario runs with increasing amounts of rainfall input of a given duration td. Time for these runs is on the ‘what if’ time axis of Fig. 1 for each actual forecast preparation time ti serving as the initial time t0 for these off-line integrations. Thus, ‘what if’ runs corresponding to a certain actual time ti use the same initial model soil water condition. The surface runoff volume produced is plotted against the volume of required rainfall of duration td (see example in Fig. 2). This plot is then used with the estimated value of threshold runoff for Section 5. K.P. Georgakakos / Journal of Hy84 the catchment to obtain the required flash flood guidance volume. Note that for each time ti and duration td there is a single surface runoff volume Rs that achieves bankfull conditions at the outlet of the stream draining the watershed (and thus a single value of threshold runoff). Fig. 2. Model relationship (solid line) between a given volume of rainfall, Ra, of duration td and the model-generated runoff Rs for a given soil moisture deficit. The relationship is used to translate the surface runoff that is just enough to cause flooding of the draining stream at the watershed outlet (called threshold runoff) to the required volume of rainfall over a given duration td (called flash flood guidance of duration td). 2. Mathematical development For the SAC model a watershed consists of three distinct areas, each with a different surface runoff production mechanism along the vertical but all hydraulically connected to the stream network (see schematic in upper panel of Fig. 3). The model generates surface runoff through the following mechanisms: (a) direct surface runoff from the permanently impervious portion of the watershed (fraction b2) produced immediately after rain starts over the watershed; (b) surface runoff from the pervious area of the watershed, produced after the upper soil water capacity is exceeded and the precipitation rate is higher than the percolation, interflow and evapotranspiration rates; (c) surface runoff from the time-variable impervious area of maximum area fraction b2 produced after the upper soil water tension requirements are met. In the latter case and under continuing rainfall, the area fraction of the additional impervious area increases to encompass more area adjacent to the streams in the manner In this work, a piecewise time-continuous version of the SAC model time-discrete equations is presented, the equations are integrated analytically to produce the requisite surface runoff, and the properties of the result are explored for apparently the first time. Links to the results of a simpler saturation-excess model are also made. The relation- ship between threshold runoff and flash flood guidance is derived and pertinent dependencies, important for operational forecasting, are outlined. The next section presents the mathematical formu- lation and the derivation of the surface runoff solutions. Included are characteristic times of surface runoff initiation and development. Section 3 examines the properties of the solutions and their sensitivity to parameters and initial conditions. Section 4 develops the relationship between threshold runoff and flash flood guidance and explores the properties of the solution as a function of soil moisture deficit and the uncertainty level of the threshold runoff estimates. Considerations for operational application are also given in this section. Concluding remarks are in drology 317 (2006) 81–103 discussed in Section 2.3. For short-time intervals of Hy K.P. Georgakakos / Journal pertinent to the flash flood problem (!12 h), surface runoff from all three watershed areas is mainly a function of the model upper soil tension water content, the upper soil free water content and the variable additional impervious area content. In this section we derive analytical expressions for the evolution of these model states from initial conditions and for significant rainfall input as required for the estimation of flash flood guidance. In addition we also specialize the formulation for a general saturation- excess model of runoff production that may be used in place of the pervious area component of the SAC model in real time flash flood applications. Before we explore the SAC model behavior it is necessary to discuss the method of formulation and Fig. 3. Schematic representation of the watershed pervious and impervious elements of the SAC model and the associated in- and out-fluxes are show drology 317 (2006) 81–103 85 solution. The operational version of the SAC model uses discrete-time non-linear mass-balance relation- ships with threshold type non-linearities to quantify water exchanges among model components. To produce analytical solutions, the time domain is subdivided in appropriate sub-intervals and these relationships are approximated by continuous-time linear ordinary differential equations valid over a certain sub-interval. While the solutions are continu- ous functions of the independent variables throughout the time domain, the derivatives involved in the differential equations are not continuous at the times where two neighboring time intervals join. In this work, the model component behavior at these characteristic times is obtained from the solution of area fractions (upper panel). The upper zone tension and free water n in the lower panel. ep xT of Hy i(t)Oep(t). For the computation of flash flood guidance through the integrations on the ‘what if’ axes of Fig. 1, the rain rate and evapotranspiration demand rate remain constant at values i and ep, and, heretofore, we will drop their dependence on time t. The SAC model fills the gravitational or free water after the tension requirements of the upper zone are met. A characteristic watershed time is the time, tT, one of the time subintervals for a one-sided limit toward the characteristic time. Because it is of interest to understand the SAC model behavior as the model components fill up (pertinent to flash flood situations), we approximate the values at the characteristic times from the solution of the earlier times sub-interval as time approaches the characteristic time from below. 2.1. Upper zone tension water The upper zone tension water element of the SAC model receives water as rain rate, i, from the atmosphere, and discharges water to the atmosphere as water vapor through evapotranspiration, eT, and, upon saturation, as excess flow, rf, to the upper zone free water element. The lower panel of Fig. 3 presents a schematic showing the upper zone elements and their in- and out-fluxes. The governing equations of the upper zone tension water content, xT, may be written in a piecewise time-continuous form as dxTðtÞ dt Z iðtÞ KeTðtÞ; xT!x0T (1a) dxTðtÞ dt Z 0; xTðtÞ Z x0T (1b) with eTðtÞ Z epðtÞ xTðtÞ x0T (2) In the previous equations, xT(t) is the upper zone tension water content at time t, x0T is the capacity of the upper zone tension water content (a model parameter), and ep(t) is the evapotranspiration demand rate. The evapotranspiration demand com- bines the atmospherically-forced potential evapor- ation with the transpiration of the watershed plant cover under saturated moisture conditions. Eq. (1b) is valid for significant precipitation rates for which: K.P. Georgakakos / Journal86 when xTðtTÞZx0T from an unsaturated initial condition t0 Z t% t!T An estimate of tT may be obtained from Eq. (3) when xTðtTÞZx0T (approaches from below) for the first time: tT Z t0 K x0T ep ln ði=epÞ K1 ði=epÞ K ðxT0=x0TÞ ( ) (4) After time tT and under continuing significant rainfall forcing (iOep), the upper zone tension water is full and its evolution is governed by Eq. (1b) with initial condition x0T. At the same time, the excess water, rf, supplies input to the upper zone free water element (Fig. 3), with rf Z i Kep (5) Analysis of the result in Eq. (4) leads to the conclusion that time tT decreases as rain rate i increases, with dependence on rain rate i that is stronger for rain rates less than 25 mm/h and for typical values of ep and x 0 T (Fig. 4). As expected, higher initial saturation ratios of the upper zone tension water ðxT0=x0TÞ yield shorter tT times for typical values of ep. For i approaching ep, tT tends to infinity. It is also noted that for iOep, the volume of rainfall of duration tTKt0 required to meet the tension require- ments of the upper soil zone, i(tTKt0), is a monotonically increasing function of i. 2.2. Upper zone free water This gravitational water element of the upper soil zone receives water from the excess of the upper zone tension water (rfZiKep) and discharges water in various forms: as interflow, uf, to the channel network; as percolation, pf, to the lower zone model storages; and, upon its saturation, as surface runoff, sf, to the at time t0. Solution of the first-order linear differential Equation (1a) with an initial condition xT0 at time t0 yields: xTðtÞ Z xT0 exp K epðt K t0Þ x0T � � C ix0T 1 Kexp K epðt K t0Þ 0 � �� � ; (3) drology 317 (2006) 81–103 channel network (see lower panel of Fig. 3). of Hy In addition, it also discharges water in the form of water vapor to the atmosphere as evapotranspiration, eF. The governing equations for the upper zone free water content, xF, may be written in a piecewise time- continuous form as: dxFðtÞ dt ZKpfðtÞ KufðtÞ KeFðtÞ; (6a) Fig. 4. Dependence of time to upper zone tension water saturation, tTKt0, on rain rate, i, for typical values of x 0 T and ep, and for two different initial contents (in % of capacity) of the upper zone tension water element at time t0. K.P. Georgakakos / Journal xTðtÞ!x0T and xFðtÞ!x0F dxFðtÞ dt Z rf KpfðtÞ KufðtÞ; xTðtÞ Z x0T and xFðtÞ!x0F (6b) dxFðtÞ dt Z 0; xTðtÞ Z x0T and xFðtÞ Z x0F (6c) with ufðtÞ Z axFðtÞ (7) pfðtÞ Z xFðtÞ x0F p0 (8) eFðtÞ Z ep 1 K xTðtÞ x0T � � ; xFðtÞO0 (9a) eFðtÞ Z 0; xFðtÞ Z 0 (9b) where a is a model parameter that determines the rate of interflow production with units of inverse time, x0F is the capacity of the upper zone free water element (another SAC model parameter), and p0 is a parameter introduced in this development to account for the component of the percolation rate that depends on the lower zone storage elements of the SAC model. As was the case for Eq. (1b), and (6c) is valid for significant rainfall amounts ðiOepCp0Cax0FÞ. For significant rainfall amounts and to simplify the integrations for a term that is small, the expression of Eq. (9) for eF is used as an approximation of the expression ðepð1KxTðtÞ=x0TÞxFðtÞ=x0FÞ used in the operational Sacramento model. This approximation will slightly decrease the content of the upper zone free water element up to the time of saturation of the upper zone tension water element in the solution of Eq. (6a). In the original SAC model formulation p0 is given as a power function of the lower zone water deficit with a low exponent (1–3). This deficit is the difference of the sum of the lower zone tension and free storage element contents from the sum of their respective capacities. In most cases, temporal changes of the lower zone tension and free water elements have significantly longer time scales than those of their upper zone counterparts. In this development and with our emphasis on short flash flood time scales (less than 12-h integration intervals), we assume that the component of percolation that depends on the lower zone deficit may be adequately approximated during the flash flood guidance integrations of the model with a constant p0 that reflects the current lower zone water storage deficit. Thus, the temporal dependence of the percolation rate pf(t) is preserved only through its dependence on the upper zone free water element content (see Eq. (8)). Upon saturation of both the upper zone tension and free water elements, surface runoff, sf(t), is produced from the pervious portion of the watershed: sfðtÞ Z i Kep Kp0 Kax0F; xTðtÞ Z x0T and xFðtÞ Z x0F (10) It is noted that the integration of Eqs. (1a) and (1b) and (6a)–(6c) yields first saturation of the tension drology 317 (2006) 81–103 87 water element and then saturation of the free water to yield tF Z tT K 1 ðp0=x0FÞCa ln ðiKepÞ=ððp0=x0FÞCaÞKx0F ðiKepÞ=ððp0=x0FÞCaÞKxFT ( ) (14) where tT is given by Eq. (4) and xFT estimated by the solution in Eq. (11) for tZtT (approaching from below). After this time, tF, the production of surface runoff begins from the pervious portion of the watershed at the rate indicated by Eq. (10). The plots in Fig. 5 show that the dependence of tF on rain rate, i, is a weak function of the initial saturation level of the upper zone free water element. They also show that the relationship of tF and i is significantly dependent on the initial saturation level of the tension water element. In all cases, dependence of tF on i is stronger for lower rainfall rates, with tF decreasing of Hydrology 317 (2006) 81–103 element because the only free water element influx, rf, exists only after the upper zone tension water has reached saturation (see Eqs. (6a) and (6b)). The governing equations of the upper zone free water content are ordinary linear differential equations amenable to analytical solutions, given an initial condition xF0 at initial time t0. The solution of Eq. (6a), valid for times in the interval (t0,tT) for which the inequalities in (6a) apply and for which xT(t) is given by Eq. (3), may be written as: xFðtÞ Z i Kep ðp0=x0FÞ Ca � � ð1 KexpfKððp0=x0FÞ CaÞðt K t0Þg CxF0 expfKððp0=x0FÞ CaÞðt K t0Þg K i KepðxT0=x0TÞ ðp0=x0FÞ Ca K ðep=x0TÞ ! ½expfKepðt K t0Þ=x0Tg KexpfKððp0=x0FÞ CaÞ !ðt K t0Þg�; t0% t! tT (11) The solution of Eq. (6b) in the interval (tT,tF), with tF(tFOtT) being the time when the upper zone free water element content first reaches capacity, may be written as follows: xFðtÞ Z i Kep ðp0=x0FÞ Ca � � ð1 KexpfKððp0=x0FÞ CaÞðt K tTÞgÞ CxFT expfKððp0=x0FÞ CaÞðt K tTÞg; tT% t! tF (12) where xFT represents the solution of Eq. (11) at time tT when the upper zone tension element first reaches its capacity. Lastly, the solution of Eq. (6c) for times tRtF, for which surface runoff occurs from the upper zone free water element, is: xFðtÞ Z x0F; tF % t (13) At time tF (the second characteristic model time for the watershed), the upper zone free water element content becomes equal to this element’s capacity ðxFðtFÞZx0FÞ, and the time difference, tFKtT, may be K.P. Georgakakos / Journal88 computed from Eq. (12) (t approaching tF from below) most significantly as i increases from 10 to 25 mm/h for typical model parameter values. For i approaching ep Cp0Cax 0 F, tF tends to infinity. As was noted for the volume required to fill the tension water element, for iOepCp0Cax 0 F, the volume i(tFKt0) required to meet the deficits of both tension and free water elements is a monotonically increasing function of i. 2.3. Additional impervious area water Under continuing rainfall, after the upper zone tension water element saturates and prior to the Fig. 5. Dependence of time to upper zone free water saturation, tFK t0, on rain rate, i, for typical values of x 0 T, x 0 F, ep, p0, and a, and for two different initial contents (in % of capacity) of the upper zone tension and free water elements at time t0. of Hy saturation of the upper zone free water element (when the entire watershed produces surface runoff), the SAC model allows for the production of additional surface runoff, simulating the expansion of the saturated areas adjacent to streams and other watershed water bodies (see schematic in upper panel of Fig. 3). A new index storage function, called the additional impervious area content is introduced to simulate the expansion of the saturated areas adjacent to the streams from a zero fraction to a maximum fraction b1. The temporal evolution of this fraction, b(t), is estimated as a quadratic function of the additional impervious area normalized by the lower zone tension water capacity of the SAC model (see schematic in Fig. 3). The additional impervious area water content, xA, over and above the upper zone tension water capacity is governed by the following equations dxAðtÞ dt ZKep 1 K xTðtÞ x0T � � xAðtÞ x0T Cx 0 L � � ; t0% t! tT (15) dxAðtÞ dt Z ði KepÞ 1 K xAðtÞ x0L 2� � ; tT% t! tF (16) and dxAðtÞ dt Z 0; tF% t (17) where x0L denotes the capacity of the Sacramento model lower zone tension water element, with xAðtÞ% x0L for all times t. For continuing rainfall forcing, the solution of Eq. (15) prior to and approaching the time of saturation of the upper zone tension water element is: xAðtÞZxA0 exp ðiKepÞðtKt0Þ=ðx0TCx0LÞ � K ðix0T=epÞKxT0 x0TCx 0 L ð1KexpfKepðtKt0Þ=x0TgÞ � ; t!tT (18) with xA0 denoting the initial condition of xA(t) at time t0. The solution of Eq. (16) for times between the time, tT, when the upper zone tension water element is saturated and approaching the time, tF, when the upper K.P. Georgakakos / Journal zone free water element is saturated is: The solutions of the upper zone tension and free water elements of the SAC model bear substantial similarity to those of typical saturation-excess soil moisture models: Z dqðtÞ dt Z iðtÞ KeðtÞ KbðtÞ (22) where Z is depth in the soil column, q(t) is the soil moisture content (vol/vol) at time t assumed uniform over depth Z, e(t) is the actual evapotranspiration rate at time t, and b(t) is baseflow rate over depth Z at time t. For the wetting phase of the ‘what if’ scenarios of Fig. 1 with significant rainfall rates i(t) and given the short time scales of the flash flood process compared to the longer time scales of the baseflow process, the actual evapotranspiration rate and the baseflow rate may be given as linear functions of the soil moisture content q(t) eðtÞ Z ep qs qðtÞ; qðtÞ%qs (23) and xAðtÞ Z x0L 1CxAT=x 0 L 1KxAT=x 0 L expf2ði KepÞðt K tTÞ=x0Lg K1 1CxAT=x 0 L 1KxAT=x 0 L expf2ði KepÞðt K tTÞ=x0Lg C1 ; tT Z t! tF (19) with xAT being the solution of Eq. (18) for tZtT (approaching from below). For continuing rainfall and times greater than tF, the solution of Eq. (17) is constant and equal to the value of xA(t) obtained from Eq. (19) for tZtF (approaching from below). The surface runoff produced by the additional impervious area water element, applicable to the impervious portion of the watershed that expands during continuous rainfall, is: saf ðtÞ Z ði KepÞ xAðtÞ x0L � �2 ; tT% t! tF (20) and saf ðtÞ Z ði KepÞ; tF % t (21) 2.4. Saturation-excess soil moisture model drology 317 (2006) 81–103 89 bðtÞ Z 3qðtÞ (24) saturation excess models with only minor modifications. 3. Properties of the solutions The analytical solutions derived earlier for the temporal evolution of the upper zone tension water element content, xT(t), the upper zone free water element content, xF(t), and the additional impervious area element content, xA(t), are depicted for two initial conditions of xF(t) in Fig. 6 (xF0Z0) and Fig. 7 Fig. 6. Temporal evolution of upper zone tension and free elements of Hydrology 317 (2006) 81–103 where ep denotes potential evapotranspiration in this case, qs is the soil saturation moisture, 3 is a constant parameter for each what if axis run of Fig. 1, and Eq. (24) is a linearization of the non-linear expression of baseflow in unsaturated soils about the initial q0 soil moisture estimate at time t0 of each ‘what if’ axis. The latter assumption is analogous to considering the Sacramento model percolation rate to be a linear function of the saturation level of upper zone free water element for the purposes of flash flood guidance integrations (see Eq. (8)). Solving as in the case of Eq. (1a) and dropping dependence of i on t (i as well as ep and 3 are assumed constant for the ‘what if’ axis integrations) we obtain: qðtÞ Z q0 exp K ep Zqs C 3 Z � � ðt K t0Þ � � C i ep qs C3 1 Kexp K ep Zqs C 3 Z � � ðt K t0Þ � �� � ; t0 Z t Z ts C t0 (25) where tsCt0 is the time when the soil just becomes saturated. In a manner analogous to that used to derive Eq. (4), we may obtain the time to saturation, ts, from Eq. (25) as follows: ts ZK 1 ep Zqs C 3 Z ln qs K i ep qs C3 q0 K i ep qs C3 8< : 9= ; (26) The behavior of the solution for ts is similar to those of Eqs. (4) and (14) for tT and tF, in that increasing rainfall rates (i) and higher initial satur- ation levels (q0/qs) yield shorter times ts. After time ts and under substantial (iOepC3qs) rainfall forcing, surface runoff s(t) is generated at the constant rate: sðtÞ Z i Kep K3qs; tO ts C t0 (27) It is noted that the saturation-excess model discussed in this section may be applied to the pervious area portion of the watershed substituting the upper zone tension and free water elements of the SAC model, and with the impervious and additional impervious area portions treated in an analogous manner as in the SAC model. For this reason, the K.P. Georgakakos / Journal90 results of the following sections may be applied to and additional impervious area element expressed as ratios of contents to capacities for typical parameter values and for iZ 5 mm/h (upper panel) and iZ15 mm/h (lower panel). Note the different time-axis scale in the two panels. of Hy K.P. Georgakakos / Journal (xF0Z0.5 x 0 F). For each initial condition case, two rain rates are examined (upper and lower panels in each of the Figures). The values of the model parameters are shown in each Figure and they are typical of mid- latitude watersheds of areas of order 1000 km2. Also, shown are the values of tT and tF for each case. In both Figs. 6 and 7, the initial level of saturation for the upper zone tension water element is 50% and the initial content of the additional impervious area element is 0. In Fig. 6 the initial level of saturation of the upper zone free water element is 0%, while in Fig. 7 it is 50%. In both Figures the upper panels correspond to the low rain rate of 5 mm/h and the lower panels correspond to the medium rain rate of Fig. 7. As in Fig. 6, but for different initial conditions of the upper zone free water element content. Note the different time-axis scale in the two panels. 15 mm/h. For the same rain rate and initial condition for xT(t), the same value of the time to saturation of the upper zone tension element, tTKt0, has been obtained. This value depends substantially on the value of rain rate, i, as discussed in Fig. 4, and in this case, for iZ5 mm/h, tTKt0Z5.15 h, while for iZ 15 mm/h, tTKt0Z1.68 h. During this time the tension water element fills with a near linear dependence on time, while evaporation, eF, interflow, uf, and percolation, pf, deplete the upper zone free water element for initial conditions greater than zero (see Eq. (6a)). The additional impervious area content, over and above the upper zone tension water element content, remains at zero during this time as prescribed by the solution in Eq. (18) with a zero initial condition. Near linear dependence of xT(t) on t is due to the small magnitude (O(10K3)) of the term ðep=x0TÞ, which allows a good linear approximation of the exponential terms in Eq. (3) for long times (tKt0): xTðtÞyxT0 C ði KepxT0=x0TÞðt K t0Þ; t0% t! tT (28) The slope of the filling curve for the upper zone tension water content in Figs. 6 and 7, normalized by the corresponding capacity, is then equal to: ½i=x0TK ðep=x0TÞðxT0=x0TÞ�. Immediately after the upper zone tension water element is filled, the upper zone free water element begins to fill as the inflow, rf, exceeds the combined outflow of evaporation, interflow and percolation. Again, due to relatively small (O(10K2)) values of ðp0=x0F CaÞ, for times (tKtT) up to order O(101), the exponential terms of the solution in Eq. (12) may be well approximated by linear functions to yield: xFðtÞyxFT C i Kep KxFT p0 x0F Ca � �� � ðt K tTÞ; tT% t! tF (29) The slope of the filling curve of the relative content of the upper zone free water element depicted in Figs. 6 and 7 is then: i Kep x0F K xFT x0F p0 x0F Ca � �� � It may be seen that the approximation is drology 317 (2006) 81–103 91 relatively good even for the case of low rainfall K.P. Georgakakos / Journal of Hydrology 317 (2006) 81–10392 and zero initial upper zone free water content (upper panel in Fig. 6) where the time interval of free water filling (tFKt0) is relatively large (w10 h); the curvature of the filling line for xF(t) in the upper panel of Fig. 6 is very slight. At the same time that the upper zone free water element commences its filling process (tZtT), the additional impervious area element content starts to increase from its initial value of zero. The increase of this element lasts for the period from tT to tF at which point the entire watershed upper zone is saturated and, for continuing rain at times greater than tF, the element content remains at its value xAFZxA(tF). Inspection of the temporal evolution of the relative content of the additional impervious area element, xAðtÞ=x0L, in Figs. 6 and 7, shows that the final value, xAF, at time tF depends significantly not only on the rain rate, i, but also on the initial level of saturation of the free water element. Higher rain rates and higher initial saturation levels of the upper zone free water element lead to lower xAF values. It is noted that in these experiments the time-varying impervious areas do not reach the maximum fraction b1 because the pervious areas become saturated before this can happen. For an initial condition, xA0Z0, Eq. (18) yields a constant solution of zero for the additional impervious area content up to and including time tT. As a result, for the cases in Figs. 6 and 7, the multiplier of the exponential functions in Eq. (19) is equal to one. For small values of (tKtT), Eq. (19) may be well approximated by the expression: xAðtÞ x0L y ði KepÞðt K tTÞ x0L ; tT% t! tF (30) in which the additional impervious area content is increasing linearly with time, and with the slope of the increase being ðiKepÞ=x0L for the normalized content. This is the case in all the panels of Figs. 6 and 7. As regards the behavior of the xA(t) solution for times prior to tT (Eq. (18)), the exponential term including ½epðtK t0Þ=x0T� may be approximated by a linear function and, upon substitution in Eq. (18) we obtain: xAðtÞyxA0 expfKepð1 KxT0=x0TÞðt K t0Þ=ðx0T Cx0LÞ yxA0ð1 Kepð1 KxT0=x0TÞðt K t0Þ=ðx0T Cx0LÞÞ; t! tT ð31Þ This result suggests that the depletion of any non- zero initial content of the additional impervious area element by a substantial fraction q requires time tqKt0 given by tq K t0 Z qðx0T Cx0LÞ=ðepð1 KxT0=x0TÞÞ (32) Even for zero initial xT0 the term in the right-hand side of Eq. (32) that excludes q is of order 103 h. Thus, to deplete even small fractions of the additional impervious area element under dry weather conditions requires days if not months. We now turn our attention to the surface runoff production. The additional impervious area, adjacent to the streams and other water bodies, produces surface runoff immediately after the upper zone tension water capacity is reached at time tT according to Eq. (20). This surface runoff is produced only over a small fraction of the watershed area denoted by b1 After time tF, when the upper zone free water element reaches saturation, the entire watershed area produces surface runoff. For the additional impervious portion b1 the rate is (iKep) as specified by Eq. (21), while for the pervious portion of the watershed the rate is ðiK ep Kp0Kax 0 FÞ as specified by Eq. (10). In addition and at all times, the fraction b2 of the watershed that is permanently impervious (water bodies and imper- vious watershed surfaces directly linked to the stream network) produces surface runoff (also called direct runoff) with the rate (iKep). Denoting by s(t) the total surface runoff at time t, we may write: sðtÞ Z b2ði KepÞ; t0% t! tT (33a) sðtÞ Z b2ði KepÞ Cb1ði KepÞ xAðtÞ x0L � �2 ; tT% t! tF (33b) sðtÞ Z ðb2 Cb1Þði KepÞ C ð1 Kb2 Kb1Þ ði Kep Kp0 Kax0FÞ; tF % t (33c) The sensitivity of the temporal evolution of the surface runoff, s(t), to the rain rate, i, may be zone, there is a relationship between the volume of of Hy discerned by inspecting Figs. 8 and 9 for four different rain rates from 5 to 20 mm/h. Fig. 8 is for the additional impervious area fraction of 0.05 of the watershed area, while Fig. 9 is for the unusually large Fig. 8. Surface runoff rate in mm/h as a function of time for four different values of constant rain rate and for typical parameter values of the Sacramento model. The additional impervious area fraction, b1, is 0.05 and the permanent impervious area fraction, b2, is 0.01 of the watershed area. K.P. Georgakakos / Journal additional impervious area fraction of 0.20 of the watershed area. The other parameters have been kept at nominal values as shown in the Figures. In both cases, a small fraction of the watershed area (0.01) is assumed to be permanently impervious, and both Figures are plotted with a logarithmic surface–runoff axis for clarity. The key feature of the plots in Figs. 8 and 9 is that substantial surface runoff is generated only after time tF when the upper soil zone of the entire watershed is saturated (Eq. (33c)). The impervious area surface runoff, which is constant throughout the time period, and the surface runoff from the additional impervious area generated during the time period from tT to tF are shown in Figs. 8 and 9 to be small contributions, even for the unusual case of b1Z0.20. They can, therefore, be neglected without significant influence on the estimated volume of surface runoff production. Under these circumstances the time tF for the Sacramento model is equivalent to the saturation time ts of the saturation excess model of Eq. (26). In the next section, the relationship between the flash flood rainfall, integrated over some duration td(ZtDKt0), and the volume of model-produced surface runoff (or effective rainfall) over the same duration. Generally guidance and threshold runoff is studied in the light of the previous results. 4. Flash flood guidance properties For a given soil moisture deficit in the upper soil Fig. 9. As in Fig. 8 but for the additional impervious area fraction, b1Z0.20 of the watershed area. drology 317 (2006) 81–103 93 and because of initial soil moisture deficits in the upper soil zone, the duration, tr, of significant non- zero surface runoff may be different from td(tr%td). For watersheds with a non-negligible portion of area, b2, that is permanently impervious, and a non- negligible portion of area, b1, that is variably impervious, it may be justified to use tdZtr. Furthermore, it has been operational practice to include interflow and baseflow from the Sacramento model as part of the surface runoff production (Seann Reed, personal communication) for flash flood guidance computations and this, when significant volumes are involved, may further justify using tdZtr. In most cases, however, even when some runoff is produced from time t0 to time tF, it is small compared to that produced after time tF when the upper zone free water element content reaches its capacity (see evidence of that in Figs. 8 and 9). In that case, the assumption of near-uniform rainfall over the interval C t ð1 Kb2 Kb1Þði Kep Kp0 KaxFÞdt of Hy F (35) or Rs Z ði KepÞtr K ð1 Kb2 Kb1Þðp0 Cax0FÞtr (36) from which we get: i Z ep CRs=tr C ð1 Kb2 Kb1Þðp0 Cax0FÞ (37) The total rainfall, Ra, during the period td may be tr used for the determination of threshold runoff of duration tr (e.g. unit hydrograph component in Carpenter and Georgakakos, 1999) does not hold even approximately, and it is more appropriate for maintaining the validity of threshold runoff theory to neglect the small early runoff volume, so that tr!td. Analogous arguments may be applied for saturation- excess models of the type discussed in Section 2.4. The following analysis uses this inequality. 4.1. Flash flood guidance versus threshold runoff of a given duration To relate the threshold runoff volume of a given duration tr to flash flood guidance of duration td, we can use the model-produced surface runoff to rainfall relationship of the previous section (Eq. (33c)). We simply have to select tD(Zt0Ctd) to be the sum of the time, tI, it takes to initiate surface runoff for the initial soil moisture deficit conditions, and tr: tD Z tI C tr (34) For relatively small values of the additional impervious area and the permanently impervious area fractions b1 and b2, as appropriate for natural watersheds, significant surface runoff occurs after time tF (see results in Figs. 8 and 9 and relevant discussion). It is well justified then to set: tIZtFKt0. Neglecting surface runoff from the impervious areas during the time interval from the initial time t0 to time tF, we may use Eq. (33c), to write the surface runoff volume, Rs, over time td as: Rs Z ðtD t0 sðtÞdt Z ðtFCtr tF ðb2 Cb1Þði KepÞdtðtFCtr 0 K.P. Georgakakos / Journal94 expressed as a function of i: Ra Z itd Z iðtT K t0Þ C iðtF K tTÞ C itr (38) Thus, for a given surface runoff, Rs, substitution of i from Eq. (37) in Eq. (38), produces Ra as a function of Rs. Furthermore, when Rs is the threshold runoff of a given duration tr for the watershed of interest (that is, the surface runoff that causes bankfull flow at the outlet of the stream draining the watershed), the value of Ra obtained from Eq. (38) becomes the flash flood guidance of duration td. It is important to note that because of the dependence of rain rate i on Rs (from Eq. (37)) and the dependence of tFKtT and tTKt0 on i, the flash flood guidance duration td (see Eq. (38)) will vary with Rs even when tr remains constant. It is noted that for saturation-excess models the time td consists of the time tr and the saturation time ts derived in Eq. (26). The rest of the development is analogous to that followed for the Sacramento model. The approach taken in this section for deriving a relationship between flash flood guidance and threshold runoff is convenient for operational appli- cations for which a collection of pre-computed threshold runoff estimates of given durations (e.g. 1, 3, and 6 h) exists (e.g. see Carpenter and Georgaka- kos, 1999), and the corresponding flash flood guidance values are sought. It provides also the time tF when significant surface runoff begins. However, for certain cases this approach may be inconvenient for use with observed or forecast rainfall as the flash flood guidance duration, td, is variable with the magnitude of threshold runoff. This duration may be obtained from td Z ðtFðRsÞ K t0Þ C tr (39) where Eqs. (4), (14), and (37) may be used to express the indicated dependence on Rs (see also Fig. 5 for dependence of tFKt0 on rain rate (i). The alternative approach is to consider td as given and derive the relationship between Ra and Rs as a function of td. This is discussed in the next subsection. In the remainder of this section we explore the properties of the relationship expressed by Eq. (38) for a given tr. Eqs. (4) and (14) express the times (tTKt0) and (tFKtT) as functions of rain rate i. Through Eq. (38) and auxiliary relationship (37), the total rainfall, Ra, over the period tD becomes a function of the surface runoff (effective rainfall), Rs, produced over the drology 317 (2006) 81–103 duration tr. In addition, the indicated dependence of Fig. 10. Ra versus Rs for various initial soil saturation levels and of Hydrology 317 (2006) 81–103 95 td on Rs may be computed in Eq. (39) through the computation of the time. Fig. 10 shows examples of the Ra versus Rs relationship for three different values of tr (1, 3 and 6 h) and for various degrees of saturation of the upper zone soil water elements in the SAC model. For Figure space economy, the notation rT0 ZxT0=x 0 T and rF0ZxF0=x 0 F is used. The rainfall volumes (both actual and effective) are in mm accumulated over the duration td. The analysis is done for typical SAC model parameters as shown in the Figure. In all cases, the relationship of the actual to effective rainfall (surface runoff) is critically depen- dent on the saturation level of the upper soil zone elements in the SAC model and it is weakly dependent on tr. Again the reader is reminded that for Rs being the threshold runoff, Ra becomes the flash flood guidance. The dependence of the flash flood guidance duration, td, on the threshold runoff volume Rs of duration tr and on tr is shown in Fig. 11 for two levels of upper soil zone free and tension water element saturation (50% for the upper panel and 80% for the lower panel). It is apparent that for rainfall rates lower than 60 mm td is dependent not only on tr but also on Rs. For higher thresholds runoff (Rs) values depen- dence on Rs weakens substantially. Thus, a 3-h duration threshold runoff corresponds to flash flood guidance of duration 6 h for a threshold runoff volume of about 40 mm and 50% saturation of the upper soil zone. It corresponds to a flash flood guidance of duration 4 h for threshold runoff volume of about 100 mm and for the same level of saturation of the upper soil zone. Higher soil saturation levels yield flash flood guidance with shorter duration (td) values. For instance, the analogous figures for an 80% saturation level of the upper soil zone are about 4.2 h for RsZ40 mm and 3.6 h for RsZ100 mm. It is notable that the relationship (Ra versus Rs) in Fig. 10 approaches a linear function, especially for high rainfall volumes. As Fig. 12 indicates, there is also a nearly linear relationship between Ra and the upper zone soil tension and free water contents. It is then reasonable to examine the degree to which a linear volumetric relationship of the type: Ra ZRs Cðx0T KxT0ÞCðx0F KxF0ÞKðep Cp0 Cax0FÞtr K.P. Georgakakos / Journal (40) three different tr values. of Hy K.P. Georgakakos / Journal96 approximates the analytical solutions depicted in the panels of Fig. 10. The analytical solution and the approximation of Eq. (40) are shown in the panels of Fig. 13 for conditions corresponding to those applicable to the panels of Fig. 10. For plot clarity, in Fig. 13 we only show two levels of upper zone soil saturation in each panel. The thick lines correspond to the analytical solutions while the thin lines of the same type correspond to the approximate volumetric relationship of Eq. (40). The saturation ratio notation follows that of Fig. 10. It is apparent that the approximation of Eq. (40) (thin lines) is quite good for high rainfall volumes but it produces substantially more surface runoff (effec- tive rainfall) than the analytical solution (thick line) Fig. 11. Flash flood guidance duration td (h) as a function of the threshold runoff volume Rs (mm) and its duration tr (h). drology 317 (2006) 81–103 based on Eq. (38) for lower rainfall volumes (up to about 100 mm of actual rainfall volume). When used in the context of flash flood guidance, for which the effective rainfall corresponding to threshold runoff is input to the relationship, the approximate relationship will tend to underestimate the actual rainfall needed to produce the threshold runoff. That is, it will under- estimate the flash flood guidance and it would be biased low (conservative for forecasting flash flood occurrence). The analytical solution of Eq. (38) is used next to study the propagation of error uncertainty from Rs (when considered as threshold runoff of duration tr) to Fig. 12. Actual rainfall volume Ra (mm) as a function of the initial upper zone tension and free water element deficits and for 1 h duration RsZ5 mm (upper panel) and RsZ25 mm (lower panel). Fig. 13. Analytical (thick lines) and approximate (thin lines) Ra versus Rs relationships. K.P. Georgakakos / Journal of Hy Ra (when considered as the corresponding flash flood guidance). A total of 10,000 samples of normally distributed Rs values were generated with a mean of 25 mm/h and a standard deviation of 6.25 mm/h (corresponding to 25% of the mean). In each case, Eq. (38) was used to produce the corresponding Ra. The cumulative distribution of the sample of generated Rs values and that of the sample of the computed Ra values are displayed in Fig. 14 in normal probability paper coordinates. Two cases of initial upper zone soil saturation were considered: (a) upper zone tension water element with 50% saturation and dry upper zone free water element (upper panels in Fig. 14); and (b) both upper zone tension and free water elements with 50% saturation (lower panels in Fig. 14). In all panels, the corresponding mean and standard deviation of the samples are shown. Characteristic of all panels is that the cumulative distributions of both Rs and Ra are very nearly Gaussian (straight lines on probability paper). And, while for Rs the sample was generated to be normally distributed and the plot simply validates the numerical procedure for generating the sample, for Ra this is a new and significant result. In spite the many non- linearities present in the right-hand side of Eq. (38) through non-linear dependence of times tT and tF on i and, via Eq. (37), on Rs, the normal character of the Rs distribution is largely preserved in the derived Ra. However, the coefficient of variation of the Ra is substantially smaller than that of the Rs, mainly due to the increase of the mean in the transformation Rs to Ra in which the standard deviation remains almost constant. The reduction of the percent error in the transformation of threshold runoff to flash flood guidance is dependent on the degrees of saturation of the initial upper soil zone tension and free water contents, and for the case depicted in Fig. 14 it is 16.4% (Z25–8.6%) for the conditions of the upper panels and it is 15% (Z25–10%) for the conditions of the lower panels. The larger the initial soil water deficit of the upper soil zone is, the greater the reduction in percent error is expected to be through the Rs to Ra transformation. This is confirmed by the results of Fig. 15 showing the dependence of the coefficient of variation of Ra (CV(Ra)) on the upper zone soil water element deficits for a given coefficient of variation of Rs (CV(Rs)Z drology 317 (2006) 81–103 97 0.25) and for trZ1 h. The upper panel corresponds to of Hy K.P. Georgakakos / Journal98 the nominal upper zone soil water element capacities (x0TZ50 mm; x 0 F Z20 mm), while the lower panel corresponds to increased upper zone soil water element capacities (x0TZ80 mm; x 0 F Z40 mm). Fig. 14. Cumulative distribution functions of Rs (left panels) and Ra (right cases of initial upper soil water are shown. drology 317 (2006) 81–103 In all cases 10,000 samples were drawn to generate the results in Fig. 15. While for the nominal capacities, near dry upper zone soil water elements produce a reduction of panels) (both in mm) in normal probability paper coordinates. Two zone capacity increase but also on the individual soil water element (tension versus free) increases. The influence on the CV(Ra) of parameter uncertainty in the capacities of the upper zone tension and free water elements for a fixed CV(Rs) (Z0.25) is studied next. A total of 10,000 samples of normally distributed Ra (meanZ25 mm, stdZ6.25 mm, coef. of var. Z0.25, trZ1 h) were generated. For each sample, capacities for the upper zone tension and free water elements were generated from a uniform K.P. Georgakakos / Journal of Hydrology 317 (2006) 81–103 99 the CV(Ra) of about 0.18 (Z0.25–0.07), for 70% (Z50/70!100) increased capacities near-dry elements produce a reduction of about 0.20 (Z0.25– 0.05). Dependence of the CV(Ra) on the deficit of upper soil zone tension water is more pronounced than that on the upper soil zone free water element. Numerical experiments with various capacities (results not shown) show that the reduction in coefficient of variation during the transformation Rs to Ra is dependent not only on the aggregate upper distribution with a given mean (nominal values of 50 and 20 mm for the tension and free elements, respectively) and a given range (G15 and G6 mm for tension and free, respectively). The results (shown in Fig. 16) are presented in the form of the panels shown in Fig. 15, but in this case, they incorporate both error uncertainty in Ra and parameter uncertainty in upper soil zone capacities. Characteristics of the results in Fig. 16 are: (a) CV(Ra) is substantially greater than Fig. 15. Coefficient of variation of Ra, (CV(Ra)), as a function of the deficit ratios of the upper zone tension and free water elements for normally distributed Rs errors with CV(Rs)Z0.25 and for trZ1 h. Upper panel is for nominal upper zone water element capacities and lower panel for increased upper zone water element capacities. that displayed in the upper panel of Fig. 15, especially for high soil water element deficit ratios; and (b) there is significant deviation from linear dependence on deficit ratios, especially for drier upper zone water elements; and (c) dependence of the CV(Ra) on the deficit of upper soil zone tension water is more pronounced than that of the upper soil zone free water, Fig. 16. As in the upper panel of Fig. 15, but for uncertain upper zone tension and free water element capacities. Capacities are drawn from specified uniform distributions as discussed in the text. duration tr on both the saturation ratios and the threshold runoff itself is shown in the lower panel of Fig. 17. For low threshold runoff volumes substantial difference exists between td (Z3 h) and tr (as low as 0.3 h). As Rs increases, tr increases toward td. Also, low upper zone saturation ratios (high deficit ratios) yield greater differences between tr (lower) and td. The results in Fig. 17 have significant implications for operational application when the flash flood duration, td, is specified and when (as is commonly the case) tdstr. For a certain watershed, threshold runoff, Rs, is a function of the watershed surface geomorphologic characteristics and channel geome- Fig. 17. Ra versus Rs (upper panel) and tr versus Rs (power panel) for three different upper soil zone tension and free water element saturation levels for tdZ3 h. of Hy especially for drier free water and wetter tension water elements. 4.2. Flash flood guidance of a given duration versus threshold runoff Although the analysis of the previous section captures the essence of the Ra versus Rs relationship well, for operational application it is typically the duration td that is given (rather than tr). This is the case so that the flash flood guidance may be compared to observed or forecast rainfall of the same duration (td) for issuing watches and warnings. In this section, we indicate the appropriate Ra versus Rs relationship that depends on td, and indicate its properties via sensitivity analysis. It is noted that in this case, for a given td, the duration, tr, of the surface runoff (threshold runoff in the flash flood guidance context) is variable and must be computed to determine the appropriate threshold runoff values. Solving Eq. (39) for tr yields: tr Z td K ðtFðRsÞ K t0Þ (41) Substitution of tr in Eq. (37) yields i Z ep CRs=ðtd K ðtF K t0ÞÞ C ð1 Kb2 Kb1Þ !ðp0 Cax0FÞ (42) For given Rs, td and SAC model parameters, Eq. (42) is a non-linear algebraic equation in i because of the dependence of tFKt0(ZtFKtTCtTKt0) on i (see Eqs. (4) and (14)). The flash flood guidance Ra that corresponds to the threshold runoff Rs may be obtained from Ra Z itd (43) with i obtained from the solution of the non-linear algebraic Eq. (42) as a function of td and Rs using numerical methods. Fig. 17 exemplifies the dependence of Ra and tr on Rs for three different initial saturation levels of the upper soil zone tension and free water elements and for a flash flood guidance duration of 3 h. Linear dependence of the flash flood guidance, Ra, on threshold runoff, Rs, is observed in the upper panel of Fig. 17, with significant dependence on the initial saturation ratios of the upper zone water elements. K.P. Georgakakos / Journal100 Significant dependence of the threshold runoff drology 317 (2006) 81–103 try, but it also depends on the duration tr (i.e. through equations were used to describe the complex of Hy evolution of the non-linear soil water storage elements of the SAC model for: (a) elucidating the behavior of the elements of the upper soil zone of the model in terms of model parameters and inputs, and under conditions found during the computations of flash flood guidance; and (b) deriving and studying analytical relationships between the threshold runoff volume of duration tr and the corresponding flash flood guidance volume of duration td. Under assumptions that are mild for the flash flood guidance model integrations, analytical solutions were obtained for the piecewise continuous differen- tial equations of the model. These solutions describe the temporal evolution of the upper soil zone tension and free water elements, and of the additional impervious area element that simulates the expansion of the surface-runoff contributing areas under con- tinuing rainfall. These solutions are in terms of the its dependence on the unit hydrograph duration as described in Carpenter and Georgakakos, 1999). This dependence of Rs on tr provides one more relationship that must be invoked to determine the appropriate value and duration of threshold runoff to use for a given initial soil moisture condition of the upper soil zone. For instance, for a specified td of 3 h and for an 80% saturation level of the upper soil zone tension and free water elements, the intersection of the said independently-derived curve between Rs and tr, with the upper curve in the lower panel of Fig. 17 will provide the appropriate value and duration of the threshold runoff for this example situation. This required additional computation is a disadvantage for the approach taken in this section compared to that taken in the previous sub-section when Rs and tr were given and the Ra and td were determined. 5. Concluding remarks Flash flood guidance theory is revisited with a focus on the transformation of the threshold runoff volumes to flash flood guidance volumes in the context of the operational Sacramento soil moisture accounting (SAC) model and with results specialized also for a simpler general saturation-excess model. Piecewise time-continuous linear differential K.P. Georgakakos / Journal SAC model parameters, of the model forcing of rainfall and evapotranspiration demand, and of the initial conditions of upper soil zone contents. Two characteristic time intervals were identified in the wetting process: the time interval tTKt0 required to meet the tension water requirements of the upper soil zone, and the time interval tFKt0 required to initiate surface runoff from the pervious area of the watershed. These time intervals were obtained as analytical functions of the model parameters and input, and of the initial soil water element contents at time t0. It was shown that these time intervals are dependent on both the capacity of the upper soil zone to store water and on the relative magnitude of the rain rate as compared to the depletion rates of the upper soil zone (percolation, interflow, evapotranspiration). Analytical results for a general saturation-excess model show that for this model the time to soil saturation is equivalent to tF, and that this saturation time exhibits dependence with respect to initial saturation levels and precipitation input that is similar to that obtained for the Sacramento model. Analytical expressions were also obtained for the surface runoff from the pervious and the impervious model areas, and it was shown that, for most natural watersheds, the surface runoff from the pervious watershed area is by far the most significant surface runoff component for flash flood guidance compu- tations under significant rainfall forcing. In addition, neglecting the small surface runoff volumes for times prior to time tF of the initiation of the pervious area surface runoff, allows for a more appropriate use of this surface runoff volume as effective rainfall of uniform intensity for the production of threshold runoff involving unit hydrograph theory (e.g. Carpen- ter and Georgakakos 1999). Analytical expressions were also obtained between the rainfall volume Ra of duration td and the ensuing surface runoff volume Rs of duration tr. When Rs and tr are the threshold runoff and the corresponding duration that cause bankfull flow at the outlet of the stream draining the watershed, Ra represents the flash flood guidance of duration td. For substantial rainfall amounts, the Ra versus Rs relationship is nearly linear and depends on the duration of the Rs and on the upper soil zone water deficit. The duration of the flash flood guidance Ra is found to depend non-linearly on the threshold runoff Rs and on its duration tr with the drology 317 (2006) 81–103 101 difference between td and tr increasing for a drier tainty analysis of the flash flood guidance estimates it would be useful to extend the analysis to infiltration of Hy upper soil zone. For normally distributed error uncertainty in threshold runoff, Rs, of duration tr, near normally distributed error uncertainty for the flash flood guidance is obtained, but with a significantly lower coefficient of variation for sub- stantial upper soil zone water deficit. Additional model parameter uncertainty uniformly distributed over a given error range (30% of nominal value for upper zone water element capacities) increased the coefficient of variation of the error in flash flood guidance but the increased values were still signifi- cantly lower than the coefficient of variation of the threshold runoff error for substantial upper soil zone water deficit. We conclude that due to the model storage elements, even significant relative errors in threshold runoff are attenuated when translated to relative errors in flash flood guidance. For operational implementation of the theory developed (tdstr) two cases are distinguished: (a) the threshold runoff and its duration are given and the flash flood guidance and its duration are obtained as was done in Section 4.1; and (b) the threshold runoff volume and the duration of the flash flood guidance are specified and the flash flood guidance volume and the duration of the threshold runoff are obtained as was done in Section 4.2. There are pros and cons for both options. The first option requires the computation of the changing flash flood guidance duration in real time (as it depends on level of soil saturation), and, as a consequence, the use of a variable rainfall interval to produce the actual rainfall volume that is to be compared to flash flood guidance for issuing watches and warnings. In this case, however, one can predetermine off-line a small set of threshold runoff values of specified durations (e.g. 1, 3, and 6 h) for use in the flash flood guidance procedure. The second option does away with the variable flash flood guidance duration and one can specify a small set of fixed durations to facilitate the computation of the actual rainfall volumes to be compared to flash flood guidance in real time. In this case, however, a much larger set of pre-computed threshold runoff volumes and durations is necessary to accommodate the changing saturation level of the upper soil zone in real time (see lower panel of Fig. 15). The specific conditions of operational implementation are likely to dictate which option is more profitable to use. K.P. Georgakakos / Journal102 A possible extension of this work is to link a full excess models for application in areas when Horto- nian runoff is dominant. Acknowledgements The author wishes to acknowledge the comments and suggestions of Seann Reed of the Hydrology Laboratory of the NWS pertaining to current oper- ational practices for producing flash flood guidance for the United States. The review comments and sugges- tions of Eylon Shamir of HRC, Victor Koren of the NWS Hydrology Laboratory and an anonymous reviewer contributed to improving the clarity of the original manuscript. The research reported was sponsored by the NOAA National Weather Service Office of Hydrologic Development under Award No DG133W-03-SE-0904. The opinions expressed in this work are those of the author and do not necessarily reflect those of NOAA and of its sub-agencies. References Burnash, R.J.C., Ferral, R.L., McGuire, R.A., 1973. A generalized reported here to determine the relative benefit of additional surveys or enhanced resolution data or better soil model calibrations under given operational conditions. It may also be possible to utilize the analytical results reported here for the wetting process for improved calibration of the upper soil zone model parameters from observed high frequency and high flow data. The analytical solutions show that there are natural groupings of SAC model parameters that should be identified as a group rather than individu- ally. Examples are the characteristic ‘times’ (x0T=ep), (x0F=ðp0Cax0FÞ), and (x0L=2ðiKepÞ) for the upper zone tension water element, the upper zone free water element, and the additional impervious area element, respectively; and the characteristic ‘volumes’ (ix0T=ep) and ððiKepÞx0F=ðp0Cax0FÞÞ of the upper soil zone tension and free water elements, respectively. Lastly, scale uncertainty analysis of the threshold runoff estimates (functions of watershed geomorphological and channel geometry characteristics) to the uncer- drology 317 (2006) 81–103 streamflow simulation system: conceptual modeling for digital computers. US National Weather Service and California Department of Water Resources, Joint Federal-State River Forecast Center, Sacramento, California. 204 pages. Carpenter, T.M., Georgakakos, K.P., 1999. National threshold runoff estimation utilizing GIS in support of operational flash flood warning systems. Journal of Hydrology 224, 21–44. Duan, Q., Schaake, J., 2001. A priori estimation of land surface model parameters. In: Lakshmi, V., Albertson, J., Schaake, J. (Eds.), Land Surface Hydrology, Meteorology and Climate, Observations and Modeling Water Science and Application, vol. 4. American Geophysical Union, Washington, DC, pp. 77–94. Georgakakos, K.P., 1986. A generalized stochastic hydrometeor- ological model for flood and flash-flood forecasting. 1. Formulation. Water Resources Research 22 (13), 2083–2095. Georgakakos, K.P., 1992. Advances in forecasting flash floods, Proceedings of the CCNAA-AIT Joint Seminar on Prediction and Damage Mitigation of Meteorologically Induced Natural Disasters, 21–24 May. National Taiwan University, Taipei, Taiwan pp. 280–293. Georgakakos, K.P., 2004. Mitigating adverse hydrological impacts of storms on a global scale with high resolution, global flash flood guidance, Abstracts Volume of International Conference on Storms/AMOS-MSNZ National Conference, 5–9 July. Autralian Meteorological Society, Brisbane, Australia pp. 23– 30. Koren, V.I., Smith, M., Wang, D., Zhang, Z., 2000. Use of soil property data in the derivation of conceptual rainfall-runoff model parameters. Preprints, 15th Conference on Hydrology, Long Beach, CA, American Meteorological Society, 10–14 January 2000. Paper 2.16. Mogil, H.M., Monro, J.C., Groper, H.S., 1978. NWS’s flash flood warning and disaster preparedness programs. Bulletin of the American Meteorological Society 59, 690–699. Reed, S., Koren, V.I., Smith, M.B., Zhang, Z., Moreda, F., Seo, D.- J., and DMIP Participants, 2004. Overall distributed model intercomparison project results. Journal of Hydrology 298, 27–60. Smith, M.B., Seo, D.-J., Koren, V.I., Reed, S.M., Zhang, Z., Duan, Q., Moreda, F., Cong, S., 2004. The distributed model intercomparison project (DMIP): Motivation and experiment design. Journal of Hydrology 298, 4–26. Sweeney, T.L., 1992. Modernized areal flash flood guidance. NOAA Technical report NWS HYDRO 44, Hydrology Laboratory, National Weather Service, NOAA, Silver Spring, MD, October, 21 pp and an appendix. K.P. Georgakakos / Journal of Hydrology 317 (2006) 81–103 103 Analytical results for operational flash flood guidance Introduction Mathematical development Upper zone tension water Upper zone free water Additional impervious area water Saturation-excess soil moisture model Properties of the solutions Flash flood guidance properties Flash flood guidance versus threshold runoff of a given duration Flash flood guidance of a given duration versus threshold runoff Concluding remarks Acknowledgements References