modern operational flash flood warning systems based on flash flood ...

Proceedings, International Conference on Innovation, Advances and Implementation of Flood Forecasting Technology 9-13 October 2005, Bergen-Tromsø, Norway

MODERN OPERATIONAL FLASH FLOOD WARNING SYSTEMS BASED ON FLASH FLOOD GUIDANCE THEORY: PERFORMANCE EVALUATION Konstantine P. Georgakakos Hydrologic Research Center, 12780 High Bluff Drive, Suite 250, San Diego, CA 92130, USA Tel: +1 (858) 794-2726; Fax: +1 (858) 792-2519; E-mail: [email protected]

Abstract Operational flash flood warnings over large areas with high spatial and temporal resolution have become feasible with (a) the development of approaches for the extraction of useful information from radar and satellite remotely sensed data using automated recording raingauge sensors, (b) the availability of high resolution digital spatial data and the ability to derive useful hydrologic information through geographic information systems applications, (c) advances in computer technology for fast processing of data over large areas with high resolution and with the inclusion of uncertainty propagation computations, and (d) the availability of communication networks and the internet for quick dissemination of large volumes of data in the form of digital maps for floodresponse agency use. The present paper focuses on flash flood warnings based on flash flood guidance estimates. These estimates are indices of the volume of rainfall of a given duration over a small catchment that is just enough to cause minor flooding at the outlet of the draining stream. They are used with estimated or forecast precipitation over the small catchment to arrive at flash flood threat indices that form the basis of decisions regarding the dissemination of warnings. The dominant source of uncertainty in these decisions is precipitation, and probabilistic models of error uncertainty may be used with Bayesian inference methods to estimate the operating characteristics (probability of detection and false alarm rate) of the flash flood warning systems. Methodological issues and results of sensitivity analysis with respect to error model parameters are discussed. Key words: Flash flood warning, GIS models, remotely sensed precipitation data, performance evaluation, decision theory

1. INTRODUCTION The term “flash flood guidance” refers to the volume of rainfall of a given duration distributed uniformly over a small catchment that is just enough to cause minor flooding at the outlet of the draining stream. Flash flood guidance values determined statistically or based on geomorphological principles have been used in an operational environment for quickly assessing localized flash flood threats within a large area by comparing to same-duration observed or forecast rainfall accumulations (e.g., Mogil et al. 1978; Sweeney et al. 1992). In operational practice, typical durations do not exceed 6 hours and the assessments are used to produce early warnings for flash floods. A recent example of an operational implementation is the Central America Flash Flood Guidance (CAFFG) system, which uses satellite precipitation estimates and has been in operation wince July 2004 serving the areas of the seven Central American countries (~ 500,000 km2) with a 200 km2 resolution (Sperfslage et al. 2004). The details of the formulation of the elements of geomorphologically-based flash flood guidance systems are given in Carpenter et al. (1999) and Georgakakos (2005). Ntelekos et al. (2005) examine in detail the uncertainty in models and input data. On the basis of a single wet-season independent validation of the estimates of 3-hourly flash flood threats in Central America, 62% of the time threat that was estimated did materialize, 35% of the time threat that was estimated did not materialized, and 3% of the time threat was not estimated when in fact flash floods occurred. The validation involved communication with local agencies, which report the location and timing of the flash flood events that materialized. The interested reader is referred to the aforementioned published articles for further technical information on formulation and uncertainty analysis. Real-time results

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suitable for public dissemination and information on the CAFFG system may also be found on line: http://www.hrc-web.org/CAFFG.

In the present paper we will examine the issues associated with the a priori performance evaluation of a typical flash flood warning system based on flash flood guidance estimates. We will use concepts from detection theory and Bayesian analysis to derive estimates for the probability of detection and the false alarm rate to characterize the ability of the flash flood warning system to estimate flash flood threat reliably. The warning system assumed for this development is based on the simple criterion that a flash flood threat is declared for a given catchment on the basis of a flash-flood guidance estimate of a given duration, if the observed (nowcasting) or forecast precipitation accumulated over the same duration is greater than this flash flood guidance estimate. The bulk of the uncertainty that may cause unreliable warnings originates in the estimated precipitation volume. There is also uncertainty that originates in the estimates of flash flood guidance (see Georgakakos 2005 and Ntelekos et al. 2005) but for the purposes of the following development this uncertainty will be considered negligible compared to the precipitation estimation uncertainty. The mathematical formulation is developed in section 2, followed by the results section. Concluding remarks are presented in section 4 of the paper, with references listed in section 5.

2. MATHEMATICAL FORMULATION 2.1 Probabilistic models An exponential probability density function is assumed for precipitation: 1 f P ( p ) = e − p / r ; p ≥ 0 and r > 0 r

(1)

where p denotes precipitation volume of a given duration averaged over a given catchment, and r is a parameter that characterizes precipitation variability. The mean and standard deviation of precipitation are both equal to r. In this and following presentations of probability density functions, upper case letters denote the random variable(s) while lower case letters denote the values assumed by the random variables. This model of precipitation amount is continuous. Continuous density functions are appropriate for precipitation amount in areas and time periods for which the fraction of zero precipitation is not high. The exponential model is used here for simplicity because it contains only one parameter (r). The analysis of the next section may also be done with more elaborate models. The model assumed for the estimated precipitation amount of the given duration and averaged over the given catchment is:

Z = P (1 + E ) ; Z ≥ 0 and P ≥ 0

(2)

where Z denotes estimated precipitation (measured or forecast), P denotes the true precipitation, and E denotes a random error that is assumed to follow the uniform distribution:

f E (ε ) =

1 ; c – δ ≤ ε ≤ c + δ and 1 + c > δ 2δ

(3)

with c being the mean (bias) and with δ being the half-range of the uniform distribution defined in the interval: [c - δ, c + δ]. The last inequality assures positive values for Z. This multiplicative model of estimated precipitation amount is selected for simplicity in computations while it preserves a positive Z and allows for a variety of relationships between bias c and half-range δ. The following analysis

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may also utilize more elaborate error models (e.g., Georgakakos 1992 in the context of radar rainfall estimates).

2.2 Operating characteristics 2.2.1 Definitions The target of analysis consists of the following two performance metrics (e.g., Wilks 1995):

PD = Pr ob ⎡⎣ Z ≥ p* | P ≥ p* ⎤⎦ : Probability of Detection (POD)

(4)

PF = Pr ob ⎡⎣ P < p * Z ≥ p* ⎤⎦ : False Alarm Rate (FAR)

(5)

where the events depicted within brackets are conditional events, and where p* is the value of flash flood guidance of a given duration. The flash flood guidance value, p*, is assumed to have negligible uncertainty compared to the uncertainty of estimated precipitation volume for this analysis, and it has the following relationship with the mean precipitation volume, r, of the same duration:

p* ≥ r or

p* =α ≥1 r

(6)

This implies that flash flooding is rather infrequent for the hypothetical location of interest. The analysis concerns the problem of detecting a flash flood that has occurred or it is imminent to occur (if Z is a measurement estimate) or that it is forecast to occur (if Z is a forecast volume of rainfall), using as a criterion the inequality Z ≥ p*. 2.2.2 Conditional probability and Bayes Theorem Expressions for PD and PF may be found by the application of the definition of conditional probability (PD) and Bayes Theorem for probabilities (PF) (e.g., Press 1989):

PD =

Pr ob ⎣⎡ Z ≥ p* ∩ P ≥ p* ⎤⎦ Pr ob ⎣⎡ P ≥ p* ⎦⎤

(7)

Pr ob ⎣⎡ Z ≥ p* P < p* ⎤⎦ Pr ob ⎡⎣ P < p* ⎤⎦ PF = Pr ob ⎣⎡ Z ≥ p* P ≥ p* ⎦⎤ Pr ob ⎡⎣ P ≥ p* ⎤⎦ + Pr ob ⎣⎡ Z ≥ p * P < p* ⎦⎤ Pr ob ⎡⎣ P < p * ⎤⎦

(8)

where the symbol “∩” denotes the intersection of two random events. The marginal a priori probabilities of Equations (7) and (8) may be obtained from: ∞ 1 Pr ob ⎡⎣ P ≥ p* ⎤⎦ = ∫ e − p / r dp p* r

(9)

and

[

]

[

Pr ob P < p * = 1 − Pr ob P ≥ p *

]

(10)

Integration of Equation (9) gives: Pr ob ⎡⎣ P ≥ p * ⎤⎦ = e −α ; α ≥ 1

(11)

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with Prob [P ≥ p*] being equal to 0.37 for α = 1, 0.14 for α = 2, and 0.05 for α = 3. Equation (10) then gives:

Pr ob ⎡⎣ P < p* ⎤⎦ = 1 − e −α ; α ≥ 1

(12)

The expressions in Equations (7) and (8) may be simplified by defining the joint probabilities: PJ + = Pr ob ⎡⎣ Z ≥ p* ∩ P ≥ p* ⎤⎦

(13)

PJ − = Pr ob ⎡⎣ Z ≥ p * ∩ P < p * ⎤⎦

(14)

and using Equations (7) and (8), together with the definitions of conditional probabilities as functions of joint probabilities. The results are: PD =

PF =

PJ +

(15)

e −α PJ −

(16)

PJ + + PJ −

2.2.3 Determination of joint probabilities PJ+ and PJTo determine the joint probabilities PJ+ and PJ-, we first determine the conditional probability density function f Z / P ( z / p ) . This may be achieved by an application of the derived-distributions relationship (e.g., Benjamin and Cornell 1970):

⎛z ⎞ ∂ε ( z ) f Z / P ( z / p ) = f E ⎜ − 1⎟ ; p>0 ⎝ p ⎠ ∂z

(17)

where the probability density function of E has been employed, together with Equation (2) solved for E. The absolute value of the derivative on the right hand side of Equation (17) is given by: ∂ε ( z ) 1 = ; p>0 ∂z p

(18)

and substitution of Equations (3) and (18) in Equation (17) gives:

f Z / P ( z / p) =

1 2δ p

; p (1 + c − δ ) ≤ z ≤ p (1 + c + δ ) ; p > 0 and 1 + c > δ

(19)

The joint probabilities PJ + and PJ − may now be derived by integrating the joint probability density

f Z , P ( z, p ) = f Z / P ( z / p ) f P ( p ) =

1 1 − p/r e ; p>0 2δ p r

(20)

over the appropriate domain of integration. Three cases are distinguished: (a) high error half-range with respect to the error bias, δ ≥ |c|; (b) high positive error bias with respect to the error half-range, c

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> δ; and (c) high negative error bias with respect to the error half-range: |c| > δ and c < 0. Results for these three cases are obtained in the following. For δ ≥ |c|, Figure 1 shows the domain of integration for the events ⎡⎣ Z ≥ p* ∩ P ≥ p* ⎤⎦ and ⎡⎣ Z ≥ p* ∩ P < p* ⎤⎦ .

Integration domains for the determination of the joint probabilities PJ + and PJ − for δ ≥

Figure 1

|c|. Using the shaded domain areas in Figure 1, for this case of δ ≥ |c|, the joint probabilities PJ+ and PJmay be determined by the joint density fZ,P as follows: PJ + =

p*/(1+ c −δ )

∫

p*

p (1+ c +δ )

dp ∫

dz (

p*

1 1 − p/r e )+ 2δ p r

∞

∫

p*/(1+ c −δ )

p (1+ c +δ )

dp ∫

p (1+ c −δ )

dz (

1 1 − p/r e ); δ ≥| c | 2δ p r

(21)

and PJ − =

p*

∫

p*/(1+ c +δ )

p (1+ c +δ )

dp ∫

p*

dz (

1 1 − p/r e ); δ ≥| c | 2δ p r

(22)

Integration yields: PJ + =

1 + c + δ −α 1 + c − δ −α /(1+ c −δ ) p * α e − e − [ E1 (α ) − E1 ( )]; δ ≥| c | 2δ 2δ 2δ r 1+ c − δ

(23)

PJ − =

1 + c + δ −α /(1+ c +δ ) −α p* α (e −e )− [ E1 ( ) − E1 (α )]; δ ≥| c | 2δ 2δ r 1+ c + δ

(24)

and

The function E1( ) signifies the exponential integral, which may be approximated by (Abramowitz and Stegun 1970): E1 (α ) =

e −α α 2 + a1α + a2 e −α α 2 + a1α + a2 e −α [ 2 γ (α ) + ξ (α )] ≅ = α α + b1α + b2 α α 2 + b1α + b2 α

5

(25)


with a1=2.334733, a2=0.250621, b1=3.330657, and b2=1.681534, and with the error of approximation ξ(α) being less than 5x10-5 in absolute value. Once PJ+ and PJ- have been determined, one may determine PD and PF from Equations (15) and (16). To complete the analysis, the cases of high positive bias (c > δ > 0) and high negative bias (δ < |c| and c < 0) should be considered. The domains of integration for these cases are shown in the panels of Figure 2. In the second case of high negative bias, the event set [Z ≥ p* ∩ P < p*] is the empty set ø. In the first case of high positive bias, the event set [Z < p* ∩ P ≥ p*] is also the empty

set ø. As the probabilities of empty sets are equal to zero, their identification simplifies the computations considerably. The results for these cases follow:

Figure 2

Integration domains for the determination of the joint probabilities PJ+ and PJ-. Left panel: c > δ; right panel: (c < 0 and |c| > δ). The symbol ø signifies the empty set.

High positive bias:

PD = 1; c > δ PJ −

PF =

e + PJ − −α

PJ − =

(26) ; c>δ

(27)

α α − − 1 + c + δ − 1+αc +δ 1 α (e − e 1+ c −δ ) − [(1 + c + δ )γ ( )e 1+ c +δ − 2δ 2δ 1+ c + δ

(1 + c − δ )γ (

α 1+ c − δ

)e

−

α 1+ c −δ

]+ e

−

α 1+ c −δ

(28)

−e ; c >δ −α

High negative bias: PJ +

PD =

e −α

; | c |> δ and c < 0

(29)

PF = 0; | c |> δ and c < 0 PJ + =

(30)

α α − − 1 + c + δ − 1+αc +δ 1 α (e − e 1+ c −δ ) − [(1 + c + δ )γ ( )e 1+ c +δ − 2δ 2δ 1+ c + δ

(1 + c − δ )γ (

α 1+ c − δ

)e

−

α 1+ c −δ

]+ e

−

α 1+ c −δ

; | c |> δ and c < 0

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(31)


In the previous equations, function γ( ) is defined by (see also Equation (25)):

γ (α ) =

α 2 + a1α + a2 α 2 + b1α + b2

(32)

The solutions presented provide the means by which the performance of flash flood warning systems that are based on flash flood guidance theory may be examined as a function of the error characteristics: c (bias) and δ (half-range). In all cases, the range of feasible values of c and δ is constrained by the conditions of the integration domain (conditions of Equations (23), (24), (28), and (31)) and by the non-negativity constraint 1+c>δ (see also conditions of Equation (3)).

2.3 Links of error models to precipitation measurement In this section and under restrictive conditions we associate the precipitation error model parameters (Equations (2) and (3)) with the precipitation measurement process. The application of the results in this section is for nowcasting, when precipitation measurements are used to detect flash floods that are occurring or are imminent to occur. Consider the catchment of interest and a measurement process that produces mean areal precipitation accumulations over a given duration (corresponding to the duration of the flash flood guidance values). The accumulation estimates are based on either gridded values (radar or satellite measurements) or on point values (raingauge measurements). It is assumed that the scale of measurement (either the size of grids or the inter-raingauge distance) is such that there is no significant spatial correlation between grids or raingauge values for the accumulations of interests in flash flood warning systems. This will be the case in highly convective environments. Also, assume that the mean areal precipitation accumulation over the catchment of interest can be obtained using the arithmetic average of the measured values within the domain of the catchment (either there is good gridded coverage of the catchment or the gauges are well distributed over the catchment area and elevation range). Lastly, we assume that the error characteristics of all the gridded values or raingauge measurements within the catchment are the same. For instance, no significant range effects or elevation effects are considered. Denote by N the number of values contributing to the mean areal accumulations. Under the aforementioned conditions, the measured mean areal precipitation accumulation, MAPA, may be written as: MAPA =

1 N ∑ Yi N i =1

(33)

where Yi are the gridded or raingauge values. The expected value of the measurement, E{MAPA}, is: E{MAPA } =

1 N ∑ E{Yi } N i =1

(34)

or, given that all grid or raingauge sites are assumed to have the same error properties,

E{MAPA } = mZ

(35)

where mZ is the common measurement mean value. This mean value may be written as a function of the parameters of the measurement error model mean (bias) (Equations (2) and (3)): mZ = E{Z } = P (1 + c )

(36) 7


where P is the true value of the mean areal precipitation accumulation. This then may be solved for c: c=

mZ − P P

(37)

Analogous arguments may be applied for the computation of the variance of MAPA: E{( MAPA − mZ ) 2 } =

sZ2 N

(38)

where sZ is the standard deviation of the grid point or raingauge measurements, and where zero correlation among gridded values or raingauge values has been enforced. This then must be associated with the variance of the error model of Equations (2) and (3) to establish the link between the measurement process and the error model parameters: sZ2 δ2 = E{[ Z − P(1 + c )]2 } = P 2 3 N

(39)

where the variance of the uniform distribution is used. This results in:

δ=

3 sZ ( ) N P

(40)

Equations (37) and (40) associate the error model parameters with the measurement process (individual grid or raingauge value means and standard deviations, and number of grid or raingauge values within the catchment). The bias relationship is linear but the half-range relationship is nonlinear. Figure 3 shows this latter relationship between number of grids or raingauges and halfrange δ. For the same number of grids or raingauges, increasing (sZ/P) increases δ, and for achieving the same δ, higher (sZ/P) requires higher N.

Figure 3

Relationship between number N of grids or raingauges within the flash flood prone catchment and half-range δ for various values of the ratio sZ/P.

3. RESULTS AND DISCUSSION Figure 4 shows the operating characteristics PD and PF of a flash flood warning system as continuous functions of δ for zero, high positive and high negative bias c. For zero bias and for α = 1, high PD (greater than 0.7) and low PF (lower than 0.2) values are obtained throughout the range of δ. 8


Significant decrease in PD and increase in PF is observed for high negative and high positive bias (105% of |δ|), respectively. More infrequent flash flooding (higher values of α) degrades performance in all cases with the same qualitative relationship of performance among the zero and high bias cases. Using the results of Figure 3 in association with those of Figure 4 and for α = 1, we may state that under the conditions of analysis: (a) zero bias allows good performance in nowcasting (values for δ ≥ 0.5) from measurements of up to 6 grids or raingauges, even for high individual gauge measurement uncertainty (as few as 1 grid or raingauge for low measurement uncertainty); and (b) unbiased grid point fields of radar or satellite measurements over the catchment of interest clearly yield good performance (high N and low δ), but high bias of such fields degrades performance significantly even for low δ (high N). With respect to the latter observation, for radar grid estimates with N ≥ 7 and low grid error variance (sZ/P = 0.3) but with high negative bias (c = -1.05δ), PD = 0.75, a value which is also obtained by 1 raingauge station of low (sZ/P).

Figure 4

PD and PF of a flash flood warning system for zero bias, and high (relative to δ) negative and positive bias as indicated and for two different values of α (≥1).

Given the importance of the precipitation error bias, continuous dependence of PD and PF on the relative bias (c/δ) is shown in Figure 5 for δ = 0.4 and for two values of α. Asymmetric behavior about zero is observed with PD decreasing substantially as the bias becomes progressively more negative, and with PF increasing substantially as the bias becomes progressively more positive. The drop in PD is sharper than the rise in PF. Flash flood prone areas with lower frequencies of flash flood occurrence exaggerate the performance reduction due to bias. This for high bias cases affects PF more than PD.

Figure 6

PD and PF as functions of (c/δ) for a fixed δ (=0.4) and for two values of α.

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4. CONCLUDING REMARKS Operational flash flood warnings are often issued on the basis of estimated precipitation accumulations (observed or forecast) exceeding a corresponding threshold value (flash flood guidance). The performance characteristics of such warning systems are examined in this paper and a methodology for quantifying the operating characteristics of these systems (probability of detection and false alarm rate) is presented. The derivations used simple but flexible models for rainfall distributions and precipitation-estimation error distributions, which allow analytical results and the detailed study of the effects of error parameters on the performance indices. The key parameters considered are the bias and standard deviation of the precipitation error model and the relationship of the flash flood guidance to the mean precipitation amount. The uncertainty of the flash flood guidance values is considered negligible compared to that of the precipitation estimates. The results indicate that for low bias and in regions where flash flooding is frequent, flash flood warning based on a flash flood guidance threshold is likely to produce high probability of detection with a low false alarm rate. The results also underline the importance of bias in estimating rainfall, which for high magnitudes (positive or negative) may degrade performance substantially. Negative bias degrades the probability of detection and positive bias increases the false alarm rate. These results are also pertinent for the design of observing systems in flash flood prone areas. Additional research may sharpen the results of this paper by allowing for a mixed distribution for precipitation (Equation (1)), more specific models for precipitation estimation error (Equations (2) and (3)) for specific situations (flash flood nowcasting using radar, raingauge and satellite data), including spatial correlation in the results of section 2.3, and allowing for uncertainty in p*. For some of these extensions it may be necessary to resort to numerical Monte Carlo experimentation.

5. REFERENCES Abramowitz, M., and I. A. Stegun, (eds.) (1970). Handbook of Mathematical Functions with Formulas, Graphs, and Mathematical Tables. Dover Publications, Inc., New York, 228-233. Benjamin, J. R., and C. A. Cornell, (1970). Probability, Statistics, and Decision for Civil Engineers. McGraw Hill Book Co., New York, 100-134. Carpenter, T.M., J.A. Sperfslage, K.P. Georgakakos, T. Sweeney and D.L. Fread, (1999). National threshold runoff estimation utilizing GIS in support of operational flash flood warning systems, Journal of Hydrology, 224, 21–44. Georgakakos, K. P., (2005). Analytical results for operational flash flood guidance. Journal of Hydrology, doi:10.1016/j.jhydrol.2005.05.009 (in press). Georgakakos, K. P., (1992). Advances in forecasting flash floods. In Proceedings of the CCNAA-AIT Joint Seminar on Prediction and Damage Mitigation of Meteorologically Induced Natural Disasters, 21-24 May 1992, National Taiwan University, Taipei, Taiwan, 280-293. Mogil, H. M., Monro, J. C., and H. S. Groper, (1978). NWS's flash flood warning and disaster preparedness programs. Bulletin of the American Meteorological Society 59, 690-699. Ntelekos, A. A., Krajewski, W. F., and K. P. Georgakakos, (2005). On the uncertainties of flash flood guidance, Towards probabilistic forecasting of flash floods. Advances in Water Resources, 63 manuscript pages (in review). Press, S. J., (1989). Bayesian Statistics, Principles, Models, and Applications. John Wiley & Sons, Inc., New York, 233 pp. Sperfslage, J. A., Georgakakos, K. P., Carpenter, T. M., Shamir, E., Graham, N. E., Alfaro, R., and L. Soriano, (2004). Central America Flash Flood Guidance (CAFFG) User’s Guide. HRC Limited Distribution Report No. 21. Hydrologic Research Center, San Diego, CA, 82 pp. Sweeney, T. L., (1992). Modernized areal flash flood guidance. NOAA Technical Report NWS HYDRO 44, Hydrology Laboratory, National Weather Service, NOAA, Silver Spring, MD, 21 pp. and an appendix. Wilks, D. S., (1995). Statistical Methods in the Atmospheric Sciences, An Introduction. Academic Press, New York, 238-250.

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