





Realtime probabilistic forecasting of
flood stages ShienTsung Chen,
PaoShan 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)


  

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
nonstructural approach to flood mitigation, is required to prolong
the response time to carry out countermeasures of flood defense. A
realtime 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.
(1) 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
(2) 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 LangYang
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 sixhour 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 thirdgrade warning stage (5.8 m) at the LanYang Bridge, as
specified by the Water Resources Agency in Taiwan, represent the
target stage for discussion. In onehour forecasting, the
probability that the future flood stage exceeds 5.8 m is 0.0%, while
in sixhour forecasting, the exceedance probability is 33.8%. The
possibilities that the flood stage exceeds the thirdgrade 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
thirdgrade 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
thirdgrade 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
decisionmaker 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 decisionmaker 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.
References
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. 

  






