水位流量率定方法之不確定性分析

天然河川河道地文與水文參數的條件所導致的斷面水位流量關係即可視為 水文系統之一種，而河川流量情形會直接影響到水資源開發與河川整治之效果。

因此，瞭解水位流量率定關係所代表之水文系統為必要，而如何藉由此率定關係 配合水位站水位觀測，準確地決定流量成為水資源規劃利用及工程設計成敗之重 要關鍵。

計算模式經常應用在定量模擬水文系統的反應上，譬如本研究所採用模擬渠 道水理現象的迴水計算水理模式，雖然包括主要的基本水力理論，但反應水位流 量率定關係之水流與河道幾何條件的交互作用與影響，為非常複雜而難以完全掌 握，也就是對於考慮的物理現象與過程，由於缺乏完全的瞭解而存在若干的不確 定性，進而言之，所採用的許多經驗公式是基於某些假設與有限的資料量所建立 之結果以預測其間的數學關係，導致無法正確表達水理和水文真實世界之物理法 則，結果從模式所獲得之預測值常會偏離實際觀測值。

儘管大部份水文系統計算模式所採用定性關係為確定，但由於模式輸入值與 參數本質上之不確定性與序率性，則輸出值為具不確定性。以本研究而言，分析 採用簡化之一維迴水計算模式HEC-RAS 3.1，以下游斷面能量坡降等於平均河床

坡度計算出正常流水深y_n進而定出N-line線，實際上仍以曼寧公式計算建立流量與 水位率定關係，其中曼寧糙率係數n不能準確地定量估計，再加上資料量測誤差、

取樣不均勻及主觀意識判斷等所造成作業不確定性，尤其在河道裡相關參數為非 定值會隨時間與空間變化，即同實測底床坡降都會隨流量過程而變化，所以其不 確定性都非常明顯，需進一步分析驗證，尤其受橋墩流場影響之水理不確定性也 有待進一步加以驗證。而且即使應用相同的模式於相同的問題上，由於輸入參數 評估之主觀認定，不同的使用者可能會給定不同的參數值，諸如種種都顯示模式 或水力關係應用在水文系統分析所存在的不確定性。不確定性為所有物理過程所 與生具有的而且不能消除，但是可經由適當的研究加以消減。

顏氏等(Yen et al., 1986)指出在水文/水理分析與模擬中，不確定性可由下列但 非限制的來源所產生：

1. 自然過程的隨機性所伴隨之本質的不確定性。

2. 反應模擬模式之能力不足或表達系統真實物理行為的設計技術之不確定性。

3. 無法精確量化模式輸入參數所導致之不確定性。

4. 包括資料的測量誤差、非一致性與非齊性、資料管理及抄錄的錯誤等資料之 不確定性。

5. 包括無法在模擬與設計過程中列計的建造、生產、毀壞、維護及其他人為因 素之營運的不確定性。

水位流量關係之不確定性可區分為以下不同分類，這些誤差分類基本來自上 述顏氏所表列的五種來源：(1) 自然過程的隨機性所伴隨之本質的不確定性；(2) 知識不確定性；(3)資料之不確定性。甚如 2 至 5 項為定論地已知，代表水位流量 關係的可變性之本質的不確定性依然存在，上述 2 至 3 項經常當作所謂知識不確 定性，反應對真實過程未有適當瞭解的結果，包括在水位、流量與其相關參數間 關係公式的不適當假設、重要參數的忽略、參數的不正確定義及其他類似的誤 差，可由改進物理過程與參數的瞭解加以減低。模式不確定性(第 2 項)被認為大 部份水位流量關係誤差最大來源，如同迴圈與非唯一的率定關係所反應的。模式

參數不確定性(第 3 項)反應諸如估算的曼寧糙率係數參數的不確定性。至於資料 的不確定性已在若干研究有所探討(Carter and Anderon, 1963; Dickeson, 1967;

Herschy, 1975; Pelletier, 1988, 1990)，包括水位、流量、幾何及其他水流與渠道特 性的量測誤差、抄寫誤差及不適當的空間與時間取樣。而營運不確定性對於流量 計算的影響非在率定關係上，因此，不在水位流量關係之不確定性研究範疇，相 關水位流量關係之不確定性研究文獻則彙整如表 2-7 (Schmidt, 2002)。

大多數完整而理想的描述變量之不確定性為機率密度函數(PDF)，所以完整 的不確定性分析目的在評估支配不確定性來源的機率密度函數與統計動差等統 計特性，但是在大部分實際問題由於模式之數學複雜性一般都難以獲得真確的 PDF。如分析水位流量關係及其過程為非線性與高度複雜，如河床粗糙係數、底 床坡降、床質級配、底床沖淤、彎道、流路變遷、潮汐及水工構造物等參數與影 響因素都須配合實測資料加以綜合分析。不確定性分析實用上都採近似分析法，

各分析方法有不同程度的複雜化、計算複雜度與資料需求，一階變方估計法 (first-order variance estimation (FOVE) method)、蒙地卡羅模擬法(Monte Carlo simulation)或點估計法(point estimates (PE) method) 為實際研究所採用以進行不 確定性分析，不確定性分析之適當方法的選擇，端賴考慮的模式、計算時間、包 含的序率參數與相關於序率輸入值與參數之可用資料而定。相關可靠度分析方法 之研究文獻則彙整如表 2-8 (Schmidt, 2002)。

其中一階變方估計法(FOVE)在工程應用上經常採用，此方法利用泰勒級數 展開以估計在某選定展開點之模式輸出值的局部不確定性。其優劣點顏氏等(1986) 曾經加以詳細敘述，此外，Karmeshu 與 Lara-Rosano (1987)也曾指出當序率性參 數之不確定性微小時則 FOVE 為點估計法(PE)之特例。只要 FOVE 與 PE 兩者所 包含的計算量大約相同情況下，則 PE 法較為通用(Yen and Tung, 1993)。

至於蒙地卡羅模擬法則藉由序率參數所歸屬機率分佈之詳細解析以複製模 式輸出值，模式輸出值不確定性具良好精度之量化可由重複多次執行模式以達 成。甚而不僅可獲得模式輸出值之統計動差，而且可用於衍生大量樣本以使分佈

曲線符合實際機率密度函數。但是模式計算相當費時，以致蒙地卡羅模擬法在不 確定性分析較為不實用。在此種情況下，則 PE 法因為所需的模式推導較蒙地卡 羅模擬法為少，且對序率輸入參數僅要求首幾級動差會是較實用的方案，儘管其 具計算優勢，但 PE 法並非適用所有模式。相關可靠度分析方法應用在水資源工 程系統之研究文獻則彙整如表 2-9 (Schmidt, 2002)，以本研究而言，係利用一維迴 水計算模式 HEC-RAS 3.1 為工具衍生大量樣本探討不確定性，則採用 FOVE 與考 量序率性參數之不確定性的點估計法(PE)應可符合需求。

表 2-1 水位流量關係概念重要發展之相關文獻摘述表(A. R. Schmidt, 2002)

作者 年 敘述

Ellet 1853 Described empirical rating in United States, for Ohio River at Wheeling, West Virginia.

Humphries and Abbot 1963 Described early ratings for Mississippi River, as well as review of literature.

Noble 1899 Described development and application of rating for Cedar River, Washington.

Seddon 1900 Examined changes in ratings, developed method to estimate velocity of flood wave.

Newell 1901 Discussed graphical development of rating, use of first differences to check and smooth rating.

Murphy 1904a Discussed characteristics of good gauging location.

Follansbee 1994 Described early development of stream gauging by U. S. Geological Survey.

表 2-2 探討流量為水位函數關係之重要研究摘述表(A. R. Schmidt, 2002)

作者 年 敘述

Murphy 1904a Dicussed graphic development of rating, use of first and second differences to check and smooth rating, development of rating table.

Hanna 1905 Constructed rating by examining discharge, mean velocity, and area curves independently.

Barrows 1907 Suggested that sparate curves should be developed for area and mean velocity as function of stage.

Horton 1907 Presented Stevens method; examines importance of slope.

Schodar 1912 Suggested logarithmic plotting to develop ratings.

Grummann 1935 Developed ratings based on flow differences for different stage intervals.

表 2-3 以水位函數處理流量方法之相關技術文獻摘述表(A. R. Schmidt, 2002)

作者 年 敘述

Humphries and Abbott 1861 Reviewed literature related to understanding of discharge determination.

Hoyt and Grover 1912 Gave thorough overview of state-of-art of rating development and application.

Steward 1921 Examined rating types for different conditions; examines variable backwater and other obstacles.

Liddell 1927 Gave brief overview of methods of developing and applying streamflow ratings Corbett et al. 1943 Gave summary of U.S. Geological Survey

understanding and methods to measure stage and discharge and to develop and apply stage-discharge ratings.

Linsley et al. 1949 Gave brief history of methods of developing and applying streamflow ratings.

Boyer 1964 Gave brief history of flow measurement, overview of methods of developing and applying streamflow ratings.

World Meteorological Organization

1980 Gave summary of methods and understanding related to developing and applying stage-discharge ratings. This manual essentially duplicates of chapters from Rantz et al. (1982b)

Rantz et al. 1982b Gave summary of U.S. Geological Survey methods and understanding related to developing and applying stage-discharge ratings.

International Organization for Standardization

1983a Gave summary of factors to consider when establishing a gauging station that will use stage-discharge ratings.

International Organization for Standardization

1983b Gave summary of methods to develop, apply, and evaluate stage-discharge ratings.

Kennedy 1984 Gave summary of U.S. Geological Survey methods to develop and apply stage-discharge ratings.

Riggs 1985 Gave brief overview of methods of developing and applying streamflow ratings.

Herschy 1995 Gave summary of methods to develop, apply, and evaluate stage-discharge ratings.

Herschy 1999 Gave a brief summary of methods to develop, and apply stage-discharge ratings in context of discharge measurement.

表 2-4 以水位與定值坡降函數處理流量方法之相關技術文獻摘述表(A. R. Schmidt, 2002)

作者 年 敘述

Literature illustrating concept of ratings based on open-channel flow equations Borrows 1907 Made distinction between ratings for weir

stations and those defined by the open-channel flow equations, described slope as most important term affecting velocity

Beardsley 1907 Treated discharges as steady, uniform open-channel flow, assume S is constant and KA R varies with stage.

Stevens 1907 Treated discharge as steady, uniform open-channel flow, assume (K S ) is constant and A R varies with stage.

Horton 1907 Treated discharge as steady, uniform open-channel flow, KA SR varies with stage.

Boyer 1964 Discussed some factors that affect stage-discharge ratings and adjustments to account for these factors as a slope term in open-channel flow equations.

Herschy 1995 Discussed some factors that affect stage-discharge ratings as a slope term in open-channel flow equations.

Atabay and Knight 1999 Used Manning’s equation and steady, uniform flow to examine effect of bedform and overbank flows on ratings.

Dawdy et al. 2000 Combined Limerinos (1970) equation to estimate n with Manning’s equation to estimate coefficients for equation 1.

Calculation methods that are based on concept of open-channel flow

Murphy 1907 Treated discharges as steady, uniform open-channel flow, KA R varies with stage and measure water slope.

Bailey and Ray 1966 Presented methos to determine rating shape or extend rating based on convergence of flow profiles from open-channel flow equations.

Ervine and Baird 1982 Used Manning’s equation for steady, uniform flow and considered turbulent shear to examine effect of overbank flows on ratings.

Rantz et al. 1982b Presented methos to determine rating shape or extend rating based on open-channel flow equations.

Kennedy 1984 Presented methos to determine rating shape or extend rating based on open-channel flow equations.

表 2-5 以水位與坡降函數處理水位流量關係之單站方法相關技術文獻摘述表(A. R. Schmidt, 2002)

作者 年 敘述

Description of method application

Liddell 1927 Presented methods by Hall et al. (1915) and Jones (1916) to correct for changing stage.

Corbett et al. 1943 Presented methods by Boyer, Jones, Lewis, Wiggins to account for changing stage.

Discussed development and application of these ratings, and provided examples of these.

Linsley et al. 1949 Presented Jones (1916) and Wiggins methods and “change-in-stage” ratings to adjust for changing stage.

Boyer 1964 Presented Jones and Boyer methods to correct for changing stage.

Dickenson 1967 Presented Jones method to correct for changing stage. Suggested that the ratio of the maximum difference between the rising and falling slopes to the channel slope provides an indicator of whether the effect of changing stage is important or negligible.

Rantz et al. 1982b Presented detailed overview of methods by

Rantz et al. 1982b Presented detailed overview of methods by