Method of performing real-time correction of a water stage forecast

A method of performing real-time correction of a water stage forecast includes obtaining at least one predicted water stage of at least one time and a predicted water stage of a next time after the at least one time; obtaining at least one observed water stage of the at least one time; generating a system error of the water stage forecast according to the at least one observed water stage, the at least one predicted water stage, the predicted water stage of the next time, a Time Series method, and an Average Deviation method; utilizing a Kalman filter method to generate a random error of the water stage forecast; generating a water stage forecast correction of the next time according to the system error and the random error; and correcting a predicted water stage of the next time according to the water stage forecast correction and the predicted water stage.

BACKGROUND OF THE INVENTION

1. Field of the Invention

The present invention relates to a method of performing real-time correction of a water stage forecast, and particularly to a method of performing real-time correction of a water stage forecast that can utilize a Time Series method and a Kalman filter to correct a predicted water stage of a lead time. The proposed method for the error forecast is developed based on the forecasts of water stages at the lead time and the estimated water stages at previous time steps during a storm event as well as the associated the forecast error. It is Noted that this significantly differs from the other approaches for correcting the forecasted water stages.

2. Description of the Prior Art

In human history, floods are one of the most severe risks to human life and property. Therefore, water stage forecasts play an important and essential role in real-time water stage (e.g. rivers, lakes, and so on) management, where the water stage forecasts include flood control, flood warnings, reservoir operation and river regulation. The water stage forecasts can increase a lead time applied to a quantitative precipitation forecast (QPF) of a transformation of rainfall into runoff and travel time in main rivers catchments. Therefore, real-time water stage forecast, warning, and response systems aim to extend the lead time to people living on floodplains, so that they can take earlier action to save themselves and their property.

However, in any flood forecast system, uncertainty in the water stage forecast is caused by some factors (e.g. input uncertainty, model structure uncertainty, and parameter uncertainty). Therefore, because of the intrinsic uncertainty of meteorological forecasts, rainfall uncertainty (a type of input uncertainty) has more significant influence on the water stage forecast than other types of uncertainty (that is, the model structure uncertainty and the parameter uncertainty). In addition, the input uncertainty contributes inherent uncertainty in hydrologic and dynamic flow models that adds to the model structure uncertainty and the parameter uncertainty.

To sum up, reliability of the water stage forecast tends to decrease with increase of the lead time. That is to say, uncertainty in the water stage forecast generally increases with a lead time for implementing flood protection measures, so that the simulated and forecasted hydrographs may not perfectly fit the hydrographic measurements. Therefore, a water stage forecast provided by the prior art may not meet water stage forecast requirements for humanity.

SUMMARY OF THE INVENTION

An embodiment provides a method of performing real-time correction of a water stage forecast. The method includes obtaining at least one predicted water stage of at least one time and a predicted water stage of a next time after the at least one time, wherein a previous time of the at least one time is a current time; obtaining at least one observed water stage of the at least one time; generating a system error of the water stage forecast according to the at least one observed water stage of the at least one time, the at least one predicted water stage of the at least one time, the predicted water stage of the next time, a Time Series method, and an Average Deviation method; utilizing a Kalman filter method to generate a random error of the water stage forecast; generating a water stage forecast correction of the next time according to the system error and the random error; and correcting a predicted water stage of the next time according to the water stage forecast correction of the next time and the predicted water stage of the next time.

The present invention provides a method of real-time correction of water stage forecast. The method utilizes a Time Series method and a Kalman filter to correct at least one predicted water stage of at least one lead time to improve disadvantages of the prior art.

DETAILED DESCRIPTION

Please refer toFIG. 1.FIG. 1is a flowchart illustrating a method of performing real-time correction of a water stage forecast according to an embodiment. Detailed steps are as follows:

Step106: Utilize a Time Series method to generate a water stage error estimate of a time (t+1) according to the observed water stages Hobst, Hobst−1, . . . , Hobst−pand the predicted water stages Hpredt+1, Hpredt, Hpredt−1, . . . , Hpredt−p.

Step108: Utilize an Average Deviation method to generate an average error of the time (t+1) according to the observed water stages Hobst, Hobst−1, . . . , Hobst−pand the predicted water stages Hpredt+1, Hpredt, Hpredt−1, . . . , Hpredt−p.

Step110: Generate a system error εsumof the water stage forecast according to the water stage error estimate of the time (t+1) and the average error of the time (t+1).

Step112: Utilize a Kalman filter method to generate a random error εKFof the water stage forecast.

Step114: Generate a water stage forecast correction of the time (t+1) according to the system error εsumand the random error εKF.

Step116: Correct a predicted water stage of the time (t+1) according to the water stage forecast correction of the time (t+1) and the predicted water stage of the time (t+1), and broadcast a corrected predicted water stage of the time (t+1).

Step118: If the real-time correction of the water stage forecast is continuously performed, go to Step102; if no, go to Step120.

In step102, the predicted water stages Hpredt+1, Hpredt, Hpredt−1, . . . , Hpredt−pcan be obtained from a water stage forecast platform, where H*predis a predicted water stage at any time, t represents a current time, t+1 represents a lead time, t−1 represents a previous time, and t−p represents a previous pthtime of the current time. In step104, the observed water stages Hobst, Hobst−1, . . . , Hobst−pcan be obtained from the water stage forecast platform, where H*obsis an observed water stage at any time. In step106, the water stage error estimate of the lead time (that is, the time (t+1)) is generated by the Time Series method according to the observed water stages Hobst, Hobst−1, . . . , Hobst−p, the predicted water stages Hpredt+1, Hpredt, Hpredt−1, . . . , Hpredt−p, and equation (1):
εt=Hpredt−Hobst
εTSMt+1=f(Hpredt+1, Hpredt, Hpredt−1, . . . , Hpredt−p, εt, εt−1, . . . , εt−p)  (1)

As shown in equation (1), εTSMt+1is the water stage error estimate of the lead time generated by a Time Series model f, εtis a water stage error of the current time (that is, the time t), εt−1is a water stage error of the previous time (that is, the time (t−1)), and εt−pis a water stage error of the previous pthtime (that is, the time (t−p)) of the current time. As shown in equation (1), the Time Series Model f is first determined by a user. That is, a forecast error of the water stage can be calculated after an order of the Time Series model f is determined. In order to reduce time for correcting the predicted water stage, the first order of the Time Series models f are selected. That is to say, the Time Series models of AR(1), AR(2), MA(1), MA(2), AR(1, 1), and so on are selected, where AR is an abbreviation for Auto-Regression and MA is an abbreviation for Moving Average.

In order to solve a problem of insufficient observed water stages in the flood forecasting period, the Average Deviation method can be utilized. In step108, the average error of the time (t+1) is generated according to the observed water stages Hobst, Hobst−1, . . . , Hobst−pand the predicted water stages Hpredt+1, Hpredt, Hpredt−1, . . . , Hpredt−p, and equation (2):

As shown in equation (2), εADt+1is the average error of the time (t+1), NSPis a predetermined interval number for calculating the average error εADt+1of the time (t+1), Hpredt−n+iis a predicted water stage of a previous (n−i)thtime of the current time, and Hobst−n+iis an observed water stage of the previous (n−i)thtime of the current time, where (n−i) is an integer.

Because various Time Series models can be applied for different lead times, the present invention adopts a concept of Forecast combination to utilize correction results of AR(1), AR(2), MA(1), MA(2), and ARMA(1,1) and equation (3) to generate a water stage forecast correction Hpred,TStof the current time, where ARMA is an abbreviation for Auto-Regression Moving Average:

As shown in equation (3), Nmodelis a number of the Time Series model (that is 5), εTSM,mtis a water stage error estimate of the current time generated by an mthmode of the Time Series model f, Hpred,mtis the water stage forecast correction of the current time corresponding to the water stage error estimate εTSM,mtof the current time generated by the mthmode of the Time Series model f, and Hpred,TStis the water stage forecast correction of the current time generated by the Time Series model f.

In step110, the system error εsumof the water stage forecast is generated according to the water stage error estimate of the time (t+1) and the average error of the time (t+1), and equation (4):

As shown in equation (4), εTSM,mt+1is the water stage error estimate of the time (t+1) generated by the mthmode of the Time Series model f.

In theory, a real-time correction mode (that is, the Time Series model) of a predicted water stage based on the Time Series method can effectively reduce errors caused by uncertainty in various hydrological, physiographic, and meteorological models to significantly increase accuracy and reliability of information of the water stage forecast, but a water stage can still have abnormal variation caused by unpredictable and uncertain factors (e.g. uncertainty in an observed water stage or forecast rainfall). Therefore, the present invention further adopts the Kalman filter method to correct the predicted water stage.

In step112, the random error (an error of the Kalman filter method) is generated according to equation (5):
εKF=KKF(Hobst−Hpred,TSt)  (5)

As shown in equation (5), εKFis the random error, KKFis a Kalman gain, and Hpred,TSt(as shown in equation (3)) is the water stage forecast correction of the current time generated by the Time Series model f.

In step114, the water stage forecast correction of the time (t+1) is generated according to the system error εsum, the random error εKF, and equation (6):

As shown in equation (6), εcombt+1is the water stage forecast correction of the time (t+1).

In step116, the predicted water stage Hpredt+1of the time (t+1) is corrected according to the water stage forecast correction εcombt+1of the time (t+1), the predicted water stage Hpredt+1of the time (t+1), and equation (7) to generate the water stage forecast correction of the time (t+1):
Hcorrt+1=Hpredt+1+εcombt+1(7)

As shown in equation (7), Hcorrt+1is the water stage forecast correction of the time (t+1).

In Step118, if the user wants to obtain a water stage forecast correction of a time (t+2), step102to step116are repeated to generate the water stage forecast correction of the time (t+2).

Please refer toFIG. 2.FIG. 2is a diagram illustrating at least one water stage forecast correction through the method inFIG. 1. As shown inFIG. 2, the predicted water stages Hpredt, Hpredt−1, . . . , Hpredt−pand the observed water stages Hobst, Hobst−1, . . . , Hobst−pare utilized to generate a water stage forecast correction of a lead time (e.g. a time (t+1), a time (t+2), a time (t+3), and so on). Then, a corrected predicted water stage of the lead time can be obtained according to the water stage forecast correction of the lead time and a predicted water stage of the lead time.

To sum up, in theory, the real-time correction mode of the predicted water stage based on the Time Series method can effectively reduce errors caused by uncertainty in various hydrological, physiographic, and meteorological models to significantly increase accuracy and reliability of information of the water stage forecast. However, the water stage can still have abnormal variation caused by unpredictable and uncertain factors (e.g. uncertainty in the observed water stage or forecast rainfall). Therefore, the present invention utilizes the Time Series method and the Kalman filter to correct a predicted water stage of a lead time to improve disadvantages of the prior art.