On the Uncertainties of Flash Flood Guidance: Toward Probabilistic Forecasting of Flash Floods

1. Introduction

The effects of a flash flood are potentially dramatic and can be measured both in terms of property damages totaling millions or even billions of dollars and lost human lives. In its 2000 policy statement, the American Meteorological Society (AMS; AMS 2000) acknowledges flash floods as one of nature’s worst killers. Many recent examples underscore this claim. In a flash flood in Texas on 5 May 1995, 16 people were killed and $1 billion in property was destroyed in approximately one hour (e.g., Baeck and Smith 1998). In another Texas flash flood on 17 October 1994, the death toll was 22 people with a damage report of $2 billion (e.g., Smith et al. 2000). The 28 July 1997 flash flood in Fort Collins, Colorado, caused five deaths and resulted in $250 million of property damage (e.g., Weaver et al. 2000; Ogden et al. 2000; Petersen et al. 1999). Each year in the United States, almost 100 people die in flash-flood-related accidents and an average of $1 billion of property is destroyed (Chang 1998).

The benefits of uncertainty quantification in real-time forecasting were acknowledged by scientists almost a century ago (Cooke 1906). In the area of hydrologic and flash flood forecasting, uncertainty modeling has been discussed by several authors (e.g., Georgakakos and Hudlow 1984; Georgakakos 1992; Krzysztofowicz 1999; Ferraris et al. 2002), but much more remains to be done in research and development. Recently, the need for probabilistic forecasting in hydrology was addressed by Georgakakos and Krzysztofowicz (2001) and Krzysztofowicz (2001) as an important scientific and technological problem. Indeed, the AMS notes in its policy statement that the quantification of forecast uncertainty and probabilistic forecasting pose primary research challenges.

In this paper we discuss the operational flash flood real-time forecasting problem and focus on certain uncertainty quantification aspects of it. While much of our discussion is generally applicable to various systems, we use as illustration the Flash Flood Guidance System (FFGS) the National Weather Service (NWS) uses (together with precipitation forecasts) to issue warnings and watches of flash flood occurrence over the nation. We use simulation to illustrate our discussion using specific hydrologic conditions of several small basins in Oklahoma. The analysis shown herein pertains only to specific aspects of uncertainty of the operational (or any other alert) system, namely, the hydrologic and hydraulic components of it. The problem of uncertainty quantification of the entire FFGS requires consideration of several additional components (see discussion in results section) and is outside the scope of this paper.

Our paper is organized as follows: in the next section we outline the current status of the U.S. operational flash flood forecasting system and briefly describe how the system works. In section 3 we present our considerations on the current FFGS and discuss its general modifications that would enable operational quantification of uncertainty. In section 4 we present the conceptual basis for our proposed uncertainty quantification and propagation methodology. In section 5 we exemplify possible use of the probabilistic version of the proposed FFGS. We close in section 6 with general conclusions and suggestions on future research and development necessary to operationally transform the current system into a probabilistic one.

2. Flash Flood Guidance System current status

The National Weather Service in fulfilling its mission of protecting peoples’ lives and property from weather-related phenomena has established the FFGS for issuing flash flood warnings and watches over the United States. This operational system has been used in a purely deterministic fashion and consists of an important tool against the catastrophic effects of flash floods. In Fig. 1 we present a simplified schematic representation of the Flash Flood Guidance System as it is used today. The backbone of the FFGS is the threshold-runoff (Thresh-R) component, which computes the amount of effective rainfall of a given duration, uniformly distributed over an area that is capable of causing minor flooding (typically identified by bank-full conditions) at the outlet of the stream. Because of its definition, Thresh-R computation is a one-time task that is calculated offline. Thresh-R values are computed for all basins in which the FFGS is effective, for rainfall durations of 1, 3, and 6 h. Carpenter et al. (1999) summarizes the available methodologies for Thresh-R calculation on the basis of modern-day digital spatial data. Information regarding current operational use is given in Reed et al. (2002). We briefly discuss the procedure in the next section.

Deterministic Thresh-R values are calculated for thousands of small basins and are later interpolated to the Hydrologic Rainfall Analysis Project (HRAP; see Fulton et al. 1998) grid for each River Forecast Center (RFC). A hydrologic model is used to forecast the generation of runoff that certain rainfall volumes would create over given durations. This is then compared with the Thresh-R values. The hydrologic model used operationally by the NWS is the Sacramento Soil Moisture Accounting Model (SACSMA; e.g., Burnash 1995). Soil moisture conditions are obtained every 6 h from the RFCs where the hydrologic models are operated. For runoff equal to the value of Thresh-R, the actual rainfall that would cause bank-full conditions at the outlet is found from the rainfall–runoff curves produced by the hydrologic model for the specific time interval and initial soil moisture conditions. These runoff values are termed flash flood guidance (FFG). The rainfall–runoff curves are produced by feeding the Sacramento model with a series of rainfall inputs for the same initial conditions in a “what if” scenario mode. This way, FFG values are obtained for the gridded Thresh-R values for the current soil moisture conditions. FFG maps are operationally computed every 1, 3, and 6 h (12 and 24 h are optional) and are compared with the rainfall maps produced using data from the Weather Surveillance Radar-1988 Doppler (WSR-88D) radars of the Next-Generation Weather Radar (NEXRAD) network (e.g., Klazura and Imy 1993). If the FFG of a specific duration is less than the NEXRAD rainfall accumulation of the same duration and over the same area, then a flash flood warning (that the event is occurring or it is imminent to occur) is considered. A watch (preparedness alert) may be issued if forecast rainfall is greater than the FFG. No action is taken if the FFG is significantly larger from observed or forecast rainfall. Flash flood warnings and watches are issued by county. It is noted that the adoption of a simple threshold procedure for formulating watches and warnings on a national scale is necessitated by the very short lead times allowed by flash flood events for forecast and response activities.

3. Flash Flood Guidance System considerations

FFGS operation is greatly facilitated by the weather radar network, which provides real-time data on the occurrence and amount of rainfall in the region (e.g., Georgakakos 1992; Polger et al. 1994; Creutin and Borga 2003; Yates et al. 2001). As we briefly explained, radar-rainfall is used as the end product to which the output of the FFGS is compared in order to decide whether or not a flash flood warning should be issued for an area. However, it has also been acknowledged by many that the errors associated with radar-rainfall estimates are significant (Smith et al. 1996; Baeck and Smith 1998; Vieux and Bedient 1998; Young et al. 1999; Yates et al. 2001; Krajewski and Smith 2002). In addition and for the case of watches, very significant forecast errors are associated with the operational precipitation forecasts even for short lead times (e.g., Charba et al. 2003; Sokol 2003). Discussion of radar and forecast rainfall uncertainty, although fundamentally important for the flash-flood warning and watch problem, is beyond the scope of this paper (see also discussion pertaining to future research in section 6).

Unfortunately, error in rainfall estimation and forecasting are not the only sources of uncertainty in flash flood forecasting. The FFGS combines various sources of information and scientific disciplines: data on watershed geomorphologic characteristics, soil moisture data, hydrologic response models, hydraulic models of open-channel flow, and interpolation and synchronization techniques are only some of the components involved in the FFGS. Each component has uncertainty associated with it. Whether it is a variable determined with a geographic information system (GIS), a field measurement, or a hydrologic model, uncertainty is inherently bound to it. Thus, it is clear that uncertainty quantification and propagation are not only necessities because of the complexity of the system, but would add to the reliability of the system output.

a. Threshold-runoff considerations

Thresh-R calculation is an important component of the FFGS. It is done offline as a one-time task and it is used as the critical input to calculate the volume of rainfall of a given duration that under specified initial soil moisture conditions will cause bank-full flows at the outlet of a basin. The methodologies for calculating threshold runoff are analytically presented and discussed in Carpenter et al. (1999), and we briefly summarize them in this section.

Assuming linear response of the basins to excessive rainfall (e.g., Chow et al. 1986), Thresh-R, R, can be calculated from a condition of equality of the peak catchment runoff resulting from effective rainfall over a certain duration to the flooding flow at the catchment stream outlet. Using linear unit hydrograph theory, this can be mathematically expressed as follows:

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where Q_p is the flooding flow in m³ s⁻¹ (cms), q_pR is the peak of the unit hydrograph of duration t_R, normalized by the catchment area A in km². Solving for the value of R, we obtain

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Threshold-runoff solution thus depends on the flooding flow, Q_p, and the peak of the unit hydrograph, q_pR (for a specific duration, t_R).

Carpenter et al. (1999) proposed two approaches for the calculation of the unit hydrograph peak, q_pR, and two approaches for determination of the flooding flow, Q_p. Thus, there are four methodologies for computing Thresh-R based on (2). Each one of the two ways for calculating the flooding flow can be combined with each one of the two ways of calculating the peak of the unit hydrograph to obtain the threshold-runoff. In an operational basis, the choice of the methodology is dictated by the data availability that each method requires.

The first method of computing q_pR entails the calculation of the unit hydrograph with the use of Snyder’s empirical synthetic unit hydrograph method (e.g., Bras 1990; Chow et al. 1986). The calculation of Snyder’s empirical coefficients that are necessary in the method’s formulation requires unit hydrographs of the watersheds under study or unit hydrographs of watersheds that are in the same region and exhibit similar hydrologic behavior. The second approach is based on the Geomorphologic Unit Hydrograph (GUH) theory proposed by Rodriguez-Iturbe and Valdes (1979) and later advanced by Rodriguez-Iturbe et al. (1982a,b). The GUH method is a physically based approach that eliminates the uncertainty of Snyder’s empirical coefficients. On the other hand, implementation of this method requires data on channel cross sections or regional relationships between channel characteristics of small streams and upstream area characteristics that may not be available in some regions.

For the calculation of the flooding flow Q_p one can use the 2-yr return period flow as an approximation. While it is doubtful that the 2-yr return period flow can represent bank-full discharge in all cases, the good national coverage of the 2-yr return period flows that the U.S. Geological Survey (USGS) maintains nationwide supports its use. However, bank-full discharge values can differ from it significantly. Williams (1978), for example, found that the average 1.5 yr of bank-full discharge recurrence interval that resulted from the study of 36 streams had a wide array of individual values between 1 and 32 yr. Research conducted in North American and Australian streams by Dury et al. (1963), Hickin (1968), and Leopold (1994) reveals a general agreement of the bank-full recurrence intervals to be nearly 1.5 yr. In a more recent study concerning streams located in the Pacific Northwest (PNW), Castro and Jackson (2001) found the bank-full discharge recurrence interval mean value to be 1.4 yr, while Sweet and Geratz (2003) report recurrence intervals for the southeastern plain eco-region of less than 1 yr. These results suggest that the bank-full discharge recurrence intervals are inherently connected with physiographic, morphologic, and/or climatic factors. Thus, the 2-yr return period is not a “safe” flow for calculating bank-full discharge, especially in cases such as Thresh-R when it is used for nationwide application (e.g., Carpenter and Georgakakos 1995). In many cases the 2-yr return period flow is larger than the bank-full discharge and thus represents an overly optimistic solution.

The second approach is based on the Manning steady, uniform flow resistance equation and provides physical flow suitable for the flooding flow definition (Carpenter et al. 1999). This definition is rather conservative. The fact that the stream would fill with water does not necessarily mean that there would be a damaging flood in the area. On the other hand, the consequences of overestimating a flooding event are less severe than an underestimation.

The success of this method depends on the quality of the available cross-sectional data. For the small scales we are interested in, those parameters are not always available. Also, the Digital Elevation Model (DEM) data with nationwide coverage that exist today cannot be used to accurately extract small-stream cross-sectional data because of their low resolution. These limitations necessitate the use of regional relationships that relate channel cross-sectional parameters with other GIS-observable catchment characteristics. If the regional relationships are a product of a careful selection of upstream watershed properties, they can provide good estimates of the local cross-sectional characteristics (see Carpenter et al. 1999 and Leopold 1994). Furthermore, the USGS has already begun distributing DEM data with ⅓ arc s (almost 10 m) resolution through its seamless data distribution system (http://seamless.usgs.gov/) for the contiguous United States. These advances in remote sensing, along with the high-resolution orthoimagery currently available for many urban areas of the United States (less than 1 m), make the use of GIS a very promising technique for obtaining accurate measurements of channel characteristics in the future.

The peak of the unit hydrograph using the Snyder empirical synthetic unit hydrograph requires field data for the calibration of the empirical coefficients of the formulation. The data should come from watersheds that have similar storage and drainage capacity, otherwise the produced unit hydrograph would not adequately describe the hydrologic response of the watershed. This requirement can only be met if “observed” unit hydrographs exist for the watershed under study. In this context, by “observed” we mean unit hydrographs derived from observed local rainfall and flow data (rather than from synthetic or geomorphologic theories). However, there are very few basins with observed unit hydrographs maintained by the NWS, and most of the ones that exist are site specific and for larger basins. These comments and observations do not support the selection of the Snyder synthetic unit hydrograph for the computation of q_pR.

The GUH theory proposed by Rodriguez-Iturbe and Valdes (1979) attempted for the first time to theoretically relate the geomorphologic characteristics of a basin with the basin’s hydrologic response. This was done with the deployment of Horton’s empirical laws and a probabilistic description of an effective drop’s destiny inside the watershed’s streams. The theory was later developed by Rodriguez-Iturbe et al. (1982b) and was made independent of the catchment velocity, included in the expressions of the original formulation. The theory has been tested and proven to effectively describe the catchment’s hydrologic response (see, e.g., Rodriguez-Iturbe et al. 1982b; Bhaskar et al. 1997; Lee 1998). Consequently, it has been used in a number of hydrologic applications (e.g., Sorman and Abdulrazzak 1993; Snell and Sivapalan 1994; Zhang and Govindaraju 2003). These findings, along with the advancement in the GIS and remote sensing technology, make the GUH theory most appealing for the computation of the unit hydrograph peak in the threshold-runoff formulation.

Based on the above arguments we selected the GUH theory and the bank-full Manning steady uniform flow resistance formula for the discussion of the uncertainty analysis of the Thresh-R calculation. Below we repeat the relevant mathematical expression after Carpenter et al. (1999) for easy reference:

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where

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and

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where Q_p is the flooding flow (m³ s⁻¹), R is the threshold runoff of duration t_R (cm), A is the drainage area (km²), t_R is the duration of the effective rainfall (h), L is the length of the main stream of the watershed (km), R_L is Horton’s length ratio (dimensionless), D_b is the hydraulic depth at the stream outlet (m), S_c is the local channel slope (dimensionless), n is Manning’s roughness coefficient (dimensionless), and B_b is the channel top width at the stream outlet (m).

The mathematical formulation above suggests that the threshold-runoff R is a nonlinear function of six parameters: B_b, D_b, S_c, A, L, and R_L. Estimates of the first three parameters can be obtained using regional relationships (Carpenter et al. 1999). We present those regional relationships in Table 1 along with their regression correlation coefficient in a log–log domain for the test basins. Estimates of the drainage area, A, and of the length of the main stream, L, can be determined using GIS data. Uncertainty is described by the spread of the data around the regression line in this treatment. Additional uncertainty may exist due to the bias but this is unknowable at this point and we do not explicitly take it into consideration.

b. Hydrologic model considerations

In Fig. 2, we present a schematic representation of our simplified version of the Sacramento model used in the FFGS. For the purpose of our study, the permanently impervious area and the additional impervious area are neglected because in natural basins pervious area saturation generates most of the surface runoff and interflow volumes (the summation of which herein is defined as effective rainfall) in flooding conditions. This version of the model is a continuous approximation of the discrete model used in operations. For the integration times of the “what if” scenarios (up to 6 h), we also neglected temporal changes in the lower zone storage from initial conditions.

The effective rainfall is equal to

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The following equations are to be solved analytically:

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and

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where dx_T/dt is the rate of change of the upper zone tension water (UZTW) contents (mm h⁻¹), i is the rainfall intensity (mm h⁻¹), e_T is the evapotranspiration rate (mm h⁻¹), r_f is the rate of outflow to the upper zone free water (UZFW) (mm h⁻¹), dx_F/dt is the rate of change of the UZFW contents (mm h⁻¹), r_f is the rate of inflow to the UZFW from the UZTW (mm h⁻¹), P_c is the percolation rate to the deep soil layers (mm h⁻¹), a is a the interflow recession constant (1 h⁻¹) , and S_f is the surface runoff rate (mm h⁻¹). Refer to Georgakakos (1986) for a detailed presentation of the continuous version of the Sacramento model equations.

We approximate the model nonlinear threshold functions with continuous quadratic functions as follows:

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where e_P is the potential evapotranspiration rate (mm h⁻¹), x_T is the content of the UZTW (mm), x⁰_T is the capacity of the UZTW (mm), x_F is the content of the UZFW (mm), x⁰_F is the capacity of the UZTW (mm), and P₀ is the percolation rate at t = t₀ (mm h⁻¹).

The behavior of the operationally used model against the one we developed for the needs of this study, in terms of transfer from tension to free water, is illustrated in Fig. 3. In the discrete version of the model no water is transferred to the free water element before the needs of the upper zone tension water are completely met. In the case of our continuous model this transfer depends on the square of the ratio of the contents of the UZTW over the UZTW capacity [see Eq. (11)]. A continuous approximation of the original discontinuous function is justified from a physical standpoint if we consider that the flow to the UZFW cannot occur instantaneously over the entire watershed of interest after the UZTW capacity is met, but will gradually occur before this happens with a rate that increases as we approximate the UZTW capacity. The choice of a quadratic function was made to facilitate analytical development for the short integration times out to 6 h for the “what if” scenarios of effective rainfall volume estimation.

For a complete analytical solution of the model’s governing equations refer to Ntelekos (2004). The differential Eqs. (9) and (10) are quadratic Riccatti equations with closed form solutions. In this paper as examples of derived relationships, we only present the solution summary of the model governing equations, which is valid for the cases x_T < x⁰_T.

Tension water equations are

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with

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and

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Free water equations are

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with

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and

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Equation (8) can be written as follows:

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By solving the two integrals of (23) and by rearranging (Ntelekos 2004) we get

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with

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and

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Equation (24) may be used to produce amounts of effective rainfall, given initial soil moisture conditions that are below saturation, over a given period of time t_R. To illustrate the model performance, we need to specify the values of the variables that are assumed constant in the mathematical formulation. Those variables are the capacities of the upper zone tension and free water (x⁰_T, x⁰_F ), the initial percolation rate (P₀), the potential evapotranspiration rate (e_P), and the interflow recession constant (a). We assume values for these variables that are representative for natural catchments in the area of interest (Illinois River basin in Oklahoma) and apply these values to all eight basins studied herein. In the left plot of Fig. 4, we present a family of curves of actual versus effective rainfall for different initial soil moisture conditions of the tension water. The duration of the rainfall event is 3 h. The initial free water content, x_F0 is kept constant at 70% of its maximum (x⁰_F = 15 mm), and the initial tension water content, x_T0 varies from 80% to 100% of its maximum (x⁰_T = 100 mm). Ntelekos (2004) inverts the relationship in Eq. (24) to solve for (it_R), the flash flood guidance, when R_tR, the threshold runoff, is given for a certain soil water deficit condition and duration, t_R.

4. Flash Flood Guidance System uncertainties

We are now ready to discuss the uncertainty sources affecting the above-defined components of the FFGS. In this section we separately study the uncertainties of the two main components of the FFGS: the threshold-runoff and the hydrologic model. Our goal is, after quantifying these uncertainties, to propagate them through the system and investigate their synergistic effect while identifying the parameters mostly responsible for the overall uncertainty. Figure 5 shows the simplified version of the FFGS, the sources of uncertainty we accounted for, and the way that those are propagated into the system. To facilitate discussing our approach we introduce now a specific location of interest.

b. Threshold-runoff uncertainties

Calculation of threshold runoff requires six variables as we previously defined them (A, L, R_L, B_b, D_L, S_c). Those variables can be separated in two groups: the ones directly obtained with the use of GIS (A, L, R_L) and the ones obtained by regional relationships (B_b, D_b, S_c). We discuss these two groups of variables separately.

We begin with the GIS-determined variables. Uncertainty information is very limited in this case, to the best of the authors’ knowledge. Carpenter and Georgakakos (1993) address this issue by comparing GIS delineated watersheds and stream lengths (90-m DEM data) with manually estimated parameters from 1:24 000 scale topographic maps. They report errors that span from −13% to 10% for drainage area and from −7% to 32% for stream length. In view of lack of hard evidence as to the nature of the error distribution it is a common practice to assume either Gaussian or uniform distribution of error around the nominal value of a given variable. We decided to parameterize the error using a generalized beta distribution, which represents a flexible compromise between expressing our lack of knowledge using uniform or Gaussian distributions. Our decision for a beta distribution was based on three reasons: 1) to avoid the discontinuities of the uniform distribution on the extremes, 2) to sustain the uniformity and symmetry of the error since no information to suggest the opposite existed, and 3) to assign slightly higher probability to the values that are likely to be close to our best estimates.

The family of beta distributions has the following general form of probability density function:

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where

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is the incomplete beta function. For further information on the beta function, refer to Johnson et al. (1994). The beta function is completely defined by two parameters (p, q) and by defining the lower and upper limits (a, b) when it is in its general form. The generalized beta function is limited from a to b, and its pdf is symmetric when p = q. When the parameters (p, q) of the beta function are close to one, the function approaches the uniform distribution. For p = q = 1, the beta function becomes uniform. For the purposes of this work we selected a generalized beta function with parameters p = q = 1.2 to simulate the error in the measurement of the three GIS-measured variables.

We also assigned nominal values of uncertainty to define the lower and upper bounds (a, b) of the generalized beta distribution for each variable, based on the results of Carpenter and Georgakakos (1993). We assumed that those nominal values of uncertainty do not depend on the scale of the measured variable and that all errors are independent. We present the nominal values of uncertainty of the Thresh-R variables coming from GIS in Table 3. For example, 15% uncertainty level in the drainage area means that the lower bound (a) of the error distribution of this variable is at the 85% of its mean value and the upper bound (b) at the 115%, respectively. We present an example of the pdf of the generalized beta function with parameters p = q = 1.2 in Fig. 7. It is actually the distribution of the error in the measurement of the drainage area of one of the basins (cataloging number 2568).

The second group comprises the variables obtained from regional relationships. The regional relationships of Oklahoma relate the top width (B_b), the hydraulic depth (D_b), and the local channel slope (S_c) with the drainage area (A) of the basin. As we show in Table 1, these are power-law-type relationships of logarithmically transformed data while the actual data were given in the study of Carpenter and Georgakakos (1995). By performing a simple logarithmic transformation and by assuming Gaussian regression error we were able to derive the error distribution associated with each parameter and for each basin. We present here as an example the derivation of the error distribution of the hydraulic depth for all basins.

The hydraulic depth and the drainage area are related as

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Since the data have been logarithmically transformed to perform the best linear fit, the original model used by Carpenter and Georgakakos (1995) has the form

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In Eq. (28), we assume the error parameter ɛ to be normally distributed, that is, ɛ ∼ N(μ_ɛ, σ_ɛ²). Thus, ln D_b has conditional normal distribution, given ln A, lnD_b| lnA ∼ N(lnc₁ + c₂ lnA, σ²_lnDb) = N(μ_lnDb, σ²_lnDb). This means that D_b is lognormally distributed, D_b ∼ LN(μ_Db, σ²_Db). The parameters of the lognormal distribution are related to the parameters of the normal distribution through an exponential transformation with the following relationships (e.g., Kottegoda and Rosso 1997):

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The mean of the normal distribution is known (μ_Db= lnc₁ + c₂ lnA) for each case. What remains to be calculated is the variance (σ²_lnDb) given lnA. We obtain

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The variance of this error (σ_ɛ²) can be found by using Eq. (28) and the actual data of Carpenter and Georgakakos (1995). In Fig. 8 we plot the regional relationships of Carpenter et al. (1999) along with the actual data obtained from Carpenter and Georgakakos (1995). As the authors of the latter work did to obtain the regional relationships we also excluded some points (marked with red in Fig. 8) as possible outliers before calculating the variance of the error (σ_ɛ²). As a result we obtain

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which is constant for all basins. Recall that we determine the mean of each basin from

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By substituting this to (29) and (30) we obtain the mean and the variance of the lognormal distribution of D_b, for each basin. We present the results in Table 4. In Fig. 9 we show a plot of the probability density function of the hydraulic depth of one of our basins (number 2568) to give an estimate of the uncertainty associated with the measurement of this variable. By following the same methodology we obtain the parameters of the error distribution of the other two parameters (B_b and S_c). We present the results in Tables 5 and 6, respectively.

5. Probabilistic Flash Flood Guidance System (PFFGS)

Now we are ready to assess the effects of uncertainty on flash flood guidance estimates through simulation experiments. We transform the Thresh-R and the FFG components to their probabilistic forms, and based on this we finally propose a probabilistic version of the currently deterministic system.

a. Probabilistic threshold-runoff (PThresh-R)

In section 4 we determined the uncertainties of all six parameters of the Thresh-R. We use a Monte Carlo sampling experiment to study their joint effect. We assume statistical independence of these parameters. Based on 100 000 realizations we fitted some well-studied two-parameter distributions and we performed hypothesis tests to find which describes best the Thresh-R distribution. We found that the lognormal distribution well describes the outcome of the numerical experiment for all basins studied here. In Fig. 10 we show, as an illustration, the 1-h threshold-runoff distribution of one of the basins (2568). On the same plot we show the best-fit lognormal distribution. In Table 10 we present the maximum-likelihood (ML) estimates of the parameters of the lognormal distributions of the 1-, 3-, and 6-h Thresh-R for all eight basins considered. Note that the estimates represent the parameters of an asymmetric distribution. Also note that these are the parameters, μ and σ², of the lognormal distribution with probability density function:

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6. Discussion of the results

The fundamental idea of this study is that accounting for uncertainty in flash flood forecasting is not only posed by the complexity of the problem but is also a necessity that would better assist decision makers in their highly critical and sensitive flood-related decisions. In this section we summarize the procedure and the critical assumptions we made and we discuss the results we obtained throughout our simulation study. We discuss the results of the components of the Flash Flood Guidance System in the order we presented them.

For the purposes of this study, we developed a simplified system resembling the one that the National Weather Service (NWS) uses operationally to issue flash flood warnings and watches over the United States (Flash Flood Guidance System). The system we proposed had two main differences from the system that the NWS is currently using. The first difference is the selected methodology for the calculation of the threshold-runoff. Threshold-runoff is the backbone of the Flash Flood Guidance System. Currently, the NWS calculates the values of the threshold-runoff with the use of the Snyder synthetic unit hydrograph and the 2–5-yr return period flow. In this study, we selected the GUH theory and the bank-full flow with the use of the Manning steady uniform flow resistance formula for the threshold-runoff calculation. The second difference is that the system we proposed accounts for several sources of uncertainty. We considered uncertainty in the threshold-runoff calculation methodology and in the hydrologic model’s parameters and initial states. We then propagated the different sources of uncertainty to study their joint effect on the Flash Flood Guidance System.

We analyzed the system using simulation and thus we did not consider or addressed many of the operational aspects that would complicate further the issues. Our main goal was to comment on the current system’s main components and to provide a methodology based on which the system could be transformed into a probabilistic one that accounts for uncertainty. We base our conclusions on results from eight randomly chosen basins within the Illinois River Basin in Oklahoma. Their drainage areas cover a range from about 40 to about 136 km².

The calculation of the threshold-runoff with the selected methodology demands measurement of six variables. Those variables are as follows: the drainage area, the length of the main stream of the watershed, Horton’s length ratio, the top width of the stream, the hydraulic depth, and the local channel slope at the outlet of the basin. Out of those six variables, the first three are directly obtained with the use of GIS. The last three are derived from regional relationships that relate basin characteristics (i.e., the drainage area of the basin) with the desired variables with the use of power-law-type relationships.

In the case of Oklahoma we did not have sufficient uncertainty information for the error associated with the variables obtained with the use of GIS. We thus assumed that the error in the measurement of those variables could be described by a symmetric generalized beta distribution for reasons explained above. We do not claim that the assumption of the beta distribution and of the uncertainty levels we chose is the only feasible uncertainty scenario but we do believe that they are a reasonable representation of the magnitude and the behavior of these errors. We also assumed that there is no correlation between these errors. However, we acknowledge the fact that the error in the measurement of the Horton length ratio is correlated with the error in the measurement of the length of the main stream because of the definition of the Horton length ratio. Despite this, we did not account for the correlation between the two variables for the sake of simplicity and because Carpenter et al. (1999) show that small changes in the Horton length ratio measurement do not significantly change the threshold-runoff value of a basin. Our assumption leads to a conservative assessment of the overall threshold-runoff uncertainty. Future studies could further investigate the error distribution of those variables and reveal their error structure in a more detailed and comprehensive manner.

For the variables obtained indirectly with the use of the regional relationships only limited uncertainty information existed in the case of the Oklahoma area. We used a simple mathematical transformation of the regional relationships, and by assuming Gaussian linear regression error, we derived the error distributions of all three variables. The error in those variables was found to be lognormally distributed due to the power-law type of the model of the regional relationship and the overall uncertainty was larger when compared to the error of the first group of variables (those obtained from GIS measurements). The increased variability in the case of the second group of variables (top width, hydraulic depth, and local channel slope at the stream outlet) is not necessarily reflecting an increased error in their measurement. We note that in the case of Oklahoma all three variables were obtained for power-law relationships with the drainage area serving as the only predictor. It is obvious, considering the variability in nature, that there is not a one-to-one mathematical relationship between those variables and the drainage area. The relationships have only statistical meaning and the uncertainty in those variables reflects the joint effect of natural variability, sample size, and measurement error. In this case too, we assumed that the errors are uncorrelated since there were no obvious reasons indicating the opposite.

Having information about the error distributions of all the variables involved in the threshold-runoff calculation we performed a Monte Carlo simulation to reveal the threshold-runoff distribution for all the basins we considered. We assumed that the errors in those variables were not only independent from each other but they were also independent of the spatial scale of the applications. This assumption is robust for the case of the variables coming from regional relationships conditioned on the assumption of constant variance of the regression error. For the case of the variables coming from regional relationships this assumption does not lack validity since the spatial scales we investigate are in almost the same magnitude of order, and the differences due to this effect can be neglected. In an operational deployment of this system, considerations like the ones we discussed above should be taken into account.

The threshold-runoff distribution was well described by a lognormal distribution for all the basins we considered herein. We performed hypothesis tests (Kolmogorov–Smirnov) to validate this finding. We have to note though that the lognormal distribution of the threshold-runoff is the outcome of the increased uncertainty of the lognormally distributed variables coming from regional relationships. The increased uncertainty of these variables dominated the outcome of the numerical experiment and made the threshold-runoff distribution a lognormal one. Despite this, the dominance of the uncertainty of those variables suggests an interesting result: threshold-runoff is particularly sensitive to this group of variables and a decrease in their uncertainty levels would automatically translate to a decrease to the overall threshold-runoff uncertainty.

To more comprehensively investigate the impact that small changes in the uncertainty levels of the variables or biases have on the threshold-runoff we also performed a sensitivity analysis the results of that we do not present here. For an analytical description of the results of the sensitivity experiments the interested reader is referred to Ntelekos (2004). The results of the sensitivity analysis pointed out the significant impact that four out of the six variables needed for the threshold-runoff calculation have on its distribution. We found threshold-runoff to be particularly sensitive to the top width, B_b; the hydraulic depth, D_b; the drainage area, A; and the length of the main stream, L. We also found that biases have a more profound effect for smaller spatial and larger temporal scales. In contrast, we did not notice any significant differences in the impact that changes in the uncertainty levels of the variables have on the threshold-runoff for the scales we studied.

The findings of the sensitivity analysis combined with the increased uncertainty in the variables coming from regional relationships constitute a first scientific suggestion for decreasing the threshold-runoff uncertainty in a possible deployment of this system. Improved regional relationships, especially those concerning the top width and the hydraulic depth, represent one possibility of effectively reducing the overall uncertainty of threshold-runoff. Of course, as we previously stated, the natural variability is a restricting parameter but increasing the sample size and using additional predictors are ways of improving those relationships. Aside from that, the River Forecast Centers (RFCs) can be assigned the responsibility of directly measuring the cross-sectional characteristics of the streams, or better yet, coordinating these efforts with the USGS. Measurement of stream cross-sectional characteristics can be accomplished either with surveys in the field and recording of the stream characteristics, or with the deployment of lidar technology and flights over the assigned area to obtain detailed data of the aforementioned variables. Current lidar technology provides a satisfactory level of accuracy that would decrease the overall uncertainty on threshold-runoff and assure more accurate flash flood warnings.

By developing a simplified system of the operationally used Sacramento model we were able to propagate the threshold-runoff uncertainty into the FFGS. In contrast with the threshold-runoff, the FFG distribution is not unique but depends on the initial soil moisture conditions. For illustrative purposes, we chose to present the results of two sets of initial soil moisture conditions that depicted typical cases of dry and wet conditions. We studied two uncertainty scenarios, each one under two different sets of initial soil moisture conditions. In the first scenario, we propagated the threshold-runoff uncertainty assuming that the hydrologic model’s uncertainty is negligible. In the second scenario, we also considered uncertainty in the hydrologic model. We assigned different levels of uncertainty to the hydrologic model parameters and to the initial states, accounting in the latter case for enhanced model observability under high flows. The error distribution of the variables was assumed to be uniform with the form of the symmetric generalized beta for the reasons we explained before. We noted that the error in those parameters is time dependent, depending on the history of uncertain rainfall in the baseline model and on the parameter uncertainty that causes error in the states that varies with dynamic regime (e.g., dry versus wet conditions). Future studies should involve the execution of the forecast model in an ensemble mode to effectively quantify the uncertainties of those variables. For the needs of this study though, we believe that the uncertainty levels we assumed are a plausible representation of reality.

The results we obtained suggest that flash flood guidance estimates have higher uncertainty under dry initial soil moisture conditions. The uncertainty of the threshold-runoff dominated the outcome of the numerical experiments, while the additional uncertainty in the hydrologic model had limited influence on flash flood guidance results. In the latter case, the standard deviation of the FFG distribution was increased while the effect was more profound under dry conditions where the model is more uncertain. This result highlights the need for more accurate measurements of the variables that control the threshold-runoff uncertainty (mainly those coming from regional relationships) in any application of this system. We provided a probabilistic description of the flash flood guidance by fitting lognormal distributions and by performing hypothesis tests. We found that the lognormal distributions adequately described the FFG distribution for all cases expect from a slight underestimation of the FFG during the peak of the distribution under dry conditions. We concluded that for all practical applications the lognormal distributions could be used to probabilistically describe the flash flood guidance distribution.

We chose a radar-rainfall example and calculated the 3-h radar-rainfall accumulation over one of our basins to give an illustrative example of the use of probabilistic estimates of flash flood guidance with (assumed) error-free radar rainfall estimates. Having the 3-h probabilistic FFG description we calculated the probability of the radar estimate to exceed the FFG, instead of just comparing the single FFG value with the radar estimate. This is equal to the probability of a flash flood occurring for the specific basin. Of course, those probabilities need to be validated using observations of flash flood occurrence to become meaningful. The development of real-time databases with such observations is necessary. A transformation of the full FFGS would require (a) a decision system for developing flash flood guidance warnings incorporating all the relevant uncertainties and (b) a flash flood occurrence database with long-term data that would allow reliability analysis. Neither exists and the latter would be difficult to create for several years to come. Despite this, the work presented herein is a theoretical but necessary first step in dealing with developing probabilistic flash flood guidance systems.