水文模式之參數不確定性分析

Abstract

水文過程屬於隨機現象，在水文模式建立和預測上不可避免的會存在不確定性。不確定性主要是因為水文現象在空間和時間上有很高的變異性(complexity in spatial and temporal variations)，導致在資料的蒐集與模擬過程中，會與真實現象有所差異，因而產生不確定性。不確定性在整個預測過程中主要分成三類，為資料不確定性(Data uncertainty)、模式不確定性(Model structural )和參數不確定性(Model specification uncertainty or Parameter uncertainty)，此三種不確定性會影響水文模式的表現與預測的準確度，整體的不確定性可藉由評估指標來量化，如均方根誤差(Root mean square error, RMSE)、效率係數(Coefficient of efficiency, CE)和持續係數(Coefficient of persistence, CP)。不確定性分析需要大量數據來建立，為了克服資料不足的問題，可藉由Bootstrap(拔靴法)產生與實際資料特性有關的大量數據，再進行參數推估與計算評估指標以分析不確定性。 本研究資料皆由AR(Autoregressive processes)模式繁衍而來，首先探討資料在不同變異度(σx)與模式有錯誤時，參數與評估指標(RMSE、CE與CP)不確定性表現。結果顯示，同一模式不同變異度所繁衍的資料，推估之參數結果接近，評估指標除了RMSE有差異外，CE與CP結果接近，RMSE除以σx後值也相近。此三指標在有無模式不確定下CP值差異最大，其餘兩者差異皆很小。 同一筆資料帶入不同模式，其CE與CP關係為線性正相關，比例關係與ρ1有關，當ρ1越大，以CE為y軸，CP為x軸之斜率越小，故可建立不同ρ1之CE與CP線性關係。不同資料以AR(1)模式配置，CE、CP呈二次曲線關係，兩者皆為一階相關係數ρ1的函數，本研究並提出以AR(1)模式所計算之CE與CP二次曲線為門檻值。此外本研究亦發現Bootstrap方法可用來模擬真實資料的不確定性，因此可結合Bootstrap方法與CE、CP指標，提出一可用來評估實際資料的不確定性方法。

The uncertainties in constructing hydrological model are inevitable. They are mainly resulted from the complexity in spatial and temporal variations. To describe uncertainty in modeling, the uncertainty can be categorized into data uncertainty, model structural and parameter uncertainty. The overall uncertainty can be quantified by some evaluation indices, for example, Root mean square error (RMSE), Coefficient of efficiency (CE) and Coefficient of persistence (CP). Since the analysis of uncertainty is based on numerous numerical data, the Bootstrap method is used to generate a bootstrap sample that reflects characteristics of reality, then parameters and evaluation indices are estimated and calculated to describe uncertainty. The data of this research is generated from Autoregressive process (AR) modeling. The main attributes of this research can be categorized into three. Firstly, parameters and evaluation index (RMSE, CE and CP) are described in the same model with different standard deviation (σx) and under circumstances when model poses uncertainty. The results shown the variation of data have no influence on parameter and evaluation index CE and CP inferring process. Similarly, the same effect is shown when RMSE is divided by σx. Among these three indicators, the results of CP shown the largest difference when the model poses uncertainty. The relationship between CE and CP in the same dataset is calibrated by AR(1) model which is uniquely dependent on ρ1 (the first order correlation coefficient). The slope with CE for the y-axis and CP for the x-axis is identified to be decrease as ρ1 increase. Therefore the relationship between CE and CP in different ρ1 is formulated. The relationship between CE and CP in different datasets calibrated by AR(1) model is a second order polynomial function. Both CE and CP are functions of the first order correlation coefficient. This study strongly recommends that the second order polynomial function of CE and CP in AR(1) model can be used as a threshold in assessing the performance of the model. The study also shown Bootstrap method can be applied to simulate the uncertainty of the real data. Therefore Bootstrap method, CE and CP can be collaborating combined to assess the uncertainty in real data.