臺灣博碩士論文加值系統

自然界的許多物理現象均為非線性，這些物理現象具有隨機特性。傳統使用統計方法加以分析，如ARMA模式、ARIMA模式，這些方法有其適用範圍。自從1970年Mandelbrot提出碎形理論之後，碎形理論即被廣泛地應用在各個領域，其後的學者也致力於這方面的研究，在水文方面也有大量的研究成果，其中降雨量亦是研究最大的一類，碎形理論被視為分析雨量時空特性之一強有力的工具。
過去10年多重碎形理論已受到重視。Shertzer和Lovejoy[1987，1989]利用倍增瀑布(multiplicative cascade)的概念建構雨量變異性的模式。一測度(measure)是否具多重碎形結構常用的解析方法有機率分佈多重尺度法(probability distribution / multiple scaling technique，PDMS)，與軌跡動差法(trace moment method，TM)，機率分佈多重尺度法可以餘維度函數(codimension function)、軌跡動差法可以經驗動差尺度指數函數(empirical moment scaling exponent function)描述多重碎形測度，Legendre轉換描述了餘維度函數與經驗動差尺度指數函數之關係。國外學者已經多重碎形理論應用至各個領域，降雨量的時空特性是否具多重碎形結構亦是被研究的重要一環。
因此本文的主要目的為：在於分析台灣地區北、中、南與東24個雨量站之時雨量紀錄，其資料長度平均為30年，經實際分析之結果顯示：此24個雨量站均存在餘維度函數與經驗動差尺度指數函數，此意味著台灣地區之降雨量具多重碎形結構。另外由Legendre轉換檢核計算之結果，其差異並不大，由Lavalle等人在1993年提出的式子，檢核頻譜分析法與軌跡動差法之計算結果，發現其結果差異亦不大。

The phenomena of natrure are all nonlinear. These physical processes are all random. Tranditionally, we analysed the properties of them by using statistical methods such like ARMA or ARIMA model. Statistical methods were applied in suitable application areas. Since 1970 Mandelbrot proposed fractal theory, it was widely applied in many domains. The scholars were absorbed in the study of fractal theory after 1970. There is a lot of resar-ch effort in the field of Hydrology. The rainfall is a substaintial set of the study of fractal theory in Hydrology. Fractal theory can be viewed as powerful tool of analysing the tempo-ral-spatial peculiarities.
In the last decade, multifractals have been given considerable attention. Schertzer and Lovejoy(1987, 1989) modeled the variability of rainfall by a (multiplicative) cascade. The methods, probability distribution / multiple scaling technique (PDMS) and trace moment (TM), usually used to determine whether a measure to be or not to be multifractals. To dis-cribe multifractal measure, it can be employed codimesion function with PDMS method and empirical moment scaling exponent function with TM method. The relationship between codimension function and empirical moment scaling exponent function can be constructed by Legendre transform. The foreign scholars have applied multifractal theory to many appli-cation areas especially in the study of the temporal-spatial properties of rainfall that to be or not to be multifractals.
The main objective of this paper is to study the behavior of the rainfall data, collected at 24 rainfall stations in Taiwan area, over periods up to 30 years. The results show that all rainfall data exist codimension function and empirical moment scaling exponent function. This implies that the rainfall in Taiwan is multifractals. The deviation of using Legendre transform and power spectrum are not significant.

中文摘要 i
英文摘要 ii
誌謝 iii
目錄 iv
表目錄 vi
圖目錄 vii
符號說明 ix
第一章 緒 論 1
1-1 前 言 1
1-2 文獻回顧 2
1-3 研究方法 4
1-4 本文組織 5
第二章 相關理論 7
2-1 碎形幾何學 7
2-1-1 碎形之意義 7
2-1-2 碎形維度之意義 9
2-1-3 尺度不變性 12
2-2 碎形特性之解析 13
2-2-1 容積維度之計算 13
2-2-2 資訊維度之計算 13
2-2-3 相關維度之計算 14
2-3 多重碎形 15
2-3-1 多重碎形之意義 15
2-3-2 多重碎形之特性 16
2-3-3 多重碎形測度之生成法 17
2-3-4 二項式測度 曲線之特性 21
2-3-5 多項式之多重碎形測度 23
2-3-6 多重碎形測度之Legendre轉換 24
2-3-7 結語 26
2-4 多重碎形之解析. 28
2-4-1 機率分布多重尺度法 29
2-4-2 軌跡動差法 29
2-4-3 頻譜分析法 31
第三章 雨量資料 32
3-1 資料收集 32
3-2 說明 32
第四章 分析與計算結果 40
4-1 計算方法之實際應用 40
4-1-1 機率分布多重尺度法之實際應用 40
4-1-2 軌跡動差法之實際應用 46
4-1-3 頻譜分析法之實際應用 60
4-2 結果與討論 60
第五章 結論與建議 70
5-1 結 論 70
5-2 建 議 71
參考文獻 72

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