Volume 3 Issue 3 - January 25, 2008
Real-time probabilistic forecasting of flood stages
Shien-Tsung Chen, Pao-Shan Yu*

Department of Hydraulic and Ocean Engineering
*Corresponding author:yups@ncku.edu.tw

Paper published in Journal of Hydrology, 340 (1–2), pp. 63–77 (June 2007)

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In Taiwan, typhoons and southwestern convective storms that bring heavy rainfall frequently cause severe disasters due to the flash floods. The flood stage rises shortly following the beginning of a rainstorm because catchments are steep and rainfall are of high intensity. Accordingly, a flood warning system, which is a crucial non-structural approach to flood mitigation, is required to prolong the response time to carry out countermeasures of flood defense. A real-time hydrological forecasting model capable of providing flood forecasts pertaining to the flood discharge or stage plays an essential role in flood warning practices. Hydrological forecasting can be performed deterministically or probabilistically. A deterministic forecast identifies a single estimate of a predictand (variable to be predicted), while a probabilistic forecast specifies a probability distribution function pertaining to the predictand. Flood forecasting is generally performed deterministically. However, a deterministic forecast may leave the user with an illusion of certainty, which can easily lead the user to suboptimal action (Krzysztofowicz, 2001). For a more practical purpose, an operational hydrological forecasting system is required to perform probabilistically. Restated, a hydrological forecasting model needs to be designed to quantify the predictive uncertainty, and thus perform probabilistic forecasts.

This study performed probabilistic forecasting with resort to predicting the probability distributions of the forecast error. The probabilistic stage forecasts can then be constructed by combining the error distributions with a deterministic stage forecast. The basic consideration of the proposed probabilistic scheme is that many forecasts can be acquired by adding many possible values of forecast error to the single deterministic forecast . The forecast error can be expressed as , where is an observation. Given that represents an observed value and hence its uncertainty can be neglected, the variance of the forecast corresponds to the variance of the forecast error.

This probabilistic approach can account for the total uncertainty affecting the forecast with resort to the uncertainty of forecast error. The probability distribution of the forecast is calculated from the forecast errors in the calibration process, which can be viewed as samples of the global forecast errors, conditional on the given inputs (Tamea et al., 2005). Therefore the error data from model simulation with respect to the calibration events provide the information for constructing the probability distribution of the stage forecasts.

The probability distribution of error, , is conditional on the forecasting model and the model inputs. Once this error probability distribution is known, the predictive probability distribution with respect to the stage forecast can be calculated by adding the error distribution to the deterministic single forecast. The formulation can be written as
where denotes the probability distribution of the stage.

In this study, the deterministic stage forecasting was performed by a stage forecasting model constructed using support vector regression, and the error distribution was constructed with reference to the fuzzy inference model. The defuzzification process in the fuzzy inference model was modified to produce a probability distribution, instead of a crisp value, via a basic defuzzification distributions (BADD) transformation approach (Filev & Yager, 1991). Moreover, the importance sampling technique was incorporated into the defuzzification process to smooth the derived probability functions. With these predictive probabilities of the forecast errors, the probability distributions of stage forecasts can be constructed by adding the probabilistic errors to the deterministic stage forecasts. The probabilistic stage forecasts with different percent confidence intervals and a certain stage forecast with an exceedance probability can be derived accordingly.

Figure 1.  Predictive probability distribution
The proposed approach is applied to the Lang-Yang River in Taiwan pertaining to validation events of six flash floods (another 12 floods used for model calibration). Figure 1 presents the one-, three- and six-hour predictive probability distributions calculated at the 30th hour of Event 15. The range of possible flood stages widens as the lead time increases. The predictive probability distribution in Figure 1 contains all the uncertainty information in the stage forecasting. The probability distribution assigns an exceedance probability to a specific future stage. For instance, let the third-grade warning stage (5.8 m) at the Lan-Yang Bridge, as specified by the Water Resources Agency in Taiwan, represent the target stage for discussion. In one-hour forecasting, the probability that the future flood stage exceeds 5.8 m is 0.0%, while in six-hour forecasting, the exceedance probability is 33.8%. The possibilities that the flood stage exceeds the third-grade warning stage in one to six hours are correspondingly [0.00%, 0.00%, 0.52%, 5.64%, 11.19%, 33.84%], representing a straightforward probability distribution of the time to warning stage.

Figure 2.  Probabilistic stage forecasting
Figure 2 shows the results of probabilistic stage forecasting. In Figure 2, the shaded area indicates the predicted 95% confidence region, and the hollow triangles and filled circles denote the deterministic forecasts and the observed flood stage, respectively. The deterministic forecasts do not fit well with the actual stages, while the 95% confidence region encloses all observed stages in this case. In Figure 2, the actual flood exceeds the third-grade warning stage at the 34th hour (the lead time is 4 hours), while the deterministic stage forecast goes over this warning stage at the 36th hour (the lead time is 6 hours). Probabilistic forecasting identifies a probability that the stage forecast exceeds the warning stage. The probabilistic forecasting informs the user that the probability that the flood will exceed the third-grade warning stage at the 34th hour is 5.64%. Although this probability estimate is not accurate (because the actual flood exceeds the warning stage with 100% certainty), it reminds the user that the flood possibly exceeds the warning stage.

If a decision-maker makes responses merely according to the deterministic forecasts, then some improper measures might be taken, or nothing will be done, due to the underestimation of the deterministic forecasts. Conversely, if the uncertainty is considered via probabilistic forecasts, then the decision-maker should be aware that greater floods are likely in the future, and that appropriate countermeasures against the floods should be adopted.
Probabilistic forecasting is a trend, and many subjects related to this field, both hydrological and other matters, must be studied further. For example, although the probabilistic forecasting model can provide the uncertainty relating to the predictand, its predictive uncertainty can often only be interpreted by the model builder. How to convert the technical and scientific probabilistic information into simple message that can be understood by the end user, especially the general public, is still an open question.


Filev, D. & Yager, R.R. (1991) A generalized defuzzification method via BAD distributions. International Journal of Intelligent Systems, 6, 687–697.
Krzysztofowicz, R. (2001) The case for probabilistic forecasting in hydrology. Journal of Hydrology, 249, 2–9.
Tamea, S., Laio, F. & Ridolfi, L. (2005) Probabilistic nonlinear prediction of river flows. Water Resources Research, 41(9), W09421.
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