二維總體經驗模態分解法於地形數據之應用暨二維核胞淹水模式之建立

Development of BEEMD Based Smoothing Algorithm for Topography and a 2D Storage Cell Based Flood Inundation Model

高延輝(Yen Hui Koh)

指導教授 ： 黃良雄

國立臺灣大學/工學院/土木工程學研究所/碩士(2015年)

https://doi.org/10.6342/NTU.2015.00600

摘要

因水災災害防治之需求，防災單位常透過數值模型(numerical modelling)模擬和探討水災於不同時空條件下之風險與破壞，希冀降低因災害造成之人員與財物之損傷。然而過往所使用的水利淹水模式(hydrodynamic flood simulation model)多建立於淺水流方程式(shallow water equations)，其數值計算易因地形變化劇烈而發散，導致淹水模擬僅能於大尺度地形探討災害過程，不免有失真之虞。有鑑於此，本研究提出兩面向相異之方案，一以二維總體經驗模態分解法(bi-dimensional ensemble empirical mode decomposition)平滑處理複雜之地形；二以曼寧公式(Manning’s formula)取代淺水流方程式，發展一簡化淹水模式(simplified flood simulation model)。

為能於複雜地形上應用水利淹水模式，本研究第一部份發展一二維總體經驗模態分解平滑處理法，藉經驗模態分解法能按原始地形之平滑度而層層剝離出數個內含物理意義的內建模態函數(intrinsic mode function)之特性，按水利淹水模式所能容忍之地形複雜度由粗糙度最低的內建模態函數依序加入，達到地形平滑之效果。針對混模(mode-mixing)與訊號極值不足(extrema lacking)之問題，本平滑處理法提出白噪音共軛對(white noise conjugate pair)之概念，透過在分解過程中加入一正一負之白噪音，利用其相消之特性確保平滑處理後之地形能不受雜訊誤差之影響，進而降低總體經驗模態分解法所需之組數(ensemble number)，改善過往因運算量龐大而不適用於大面積應用之困境。分解結果顯示本方法能有效地偵測地形之局部特徵並剝離分解之，確保平滑後之地形不會過度失真。而隨著白噪音強度之提高，低粗糙度之內建模態函數能更詳細地表現出大尺度地形變化，讓本平滑處理法能因應不同水利淹水模式之穩定性並彈性重構所需地形之粗糙度。

考慮部分複雜地形如懸崖經平滑處理後仍可能造成水利淹水模式數值計算不穩定，本研究第二部分建立一簡化淹水模式，利用曼寧公式取代淺水流方程式以剔除造成數值運算不穩定的高非線性微分項，使其能應用於任何複雜地形上。另考量集水區邊界多不規則，且格柵網格（gridded Digital Elevation Model）在地形呈現上易因不連續而失真，本模式於數值地形上改採不規則三角形格網(triangulated irregular network)，並視每一三角形網格為一瀦蓄核胞(storage cell)，藉蓄水量於時空之分布詮釋一淹水事件。本研究在淹水體積分配上結合曼寧公式與多流向演算法(multiple flow direction algorithm)，先透過加權平均獲得曼寧公式所需之變數並計算出每一網格的總出流量，再將之按淹水高程坡度分配流至相鄰網格的分流量，使淹水模式能兼具質量守衡與降雨逕流主要受地形高程變化影響的特徵。另，本模式亦針對都市淹水發展出地下排水(underground sewer system)和建物雨水貯留利用(rainwater harvesting system)兩系統，使模擬結果能更貼近真實淹水分布。在案例模擬上，本研究先將模式應用於莫拉克颱風期間之林邊溪流域，整體結果顯示模式在淹水範圍或深度都有不錯的表現，亦可良好地模擬出河川水位之變化。在虛擬都市的模擬裡，因建物阻擾，水流多沿邊界流至下游低地，巷子內的逕流則因人孔設置密度高，在積水壅高前就先被排入地下排水道。當下水道滿載後，巷子內的逕流因無法排除而沿建物往下游蔓延，最後完全淹沒位於下游之凹地。兩案例之模擬結果皆貼近實際的淹水過程，應可提供日後淹水模式於複雜地形之應用。

關鍵字

二維總體經驗模態分解法 ； 地形平滑化處理 ； 淹水模擬 ； 瀦蓄核胞模式 ； 多流向演算法

並列摘要

As rapid population growth and land-use changes increasingly exposed human beings to a greater degree of flood hazards, disaster prevention and mitigation projects have heavily applied numerical modelling methods to assess the risks and impacts at different temporal and spatial scales. Among the wide range of models available, 2D hydrodynamic models are probably the most common tools applied in investigating flood events, owing to its rigorous physical basis and mathematics foundation. However, the presence of highly nonlinear derivatives in the momentum equations have often found triggering numerical instabilities when the model is applied over a complex topography of high curvature variation.

In this study, two different strategies are proposed to approach the difficulty. The first method applied is to reduce the local scale curvature through topography smoothing. Here a bi-dimensional ensemble empirical mode decomposition (BEEMD) based smoothing algorithm, namely the Fast and Adaptive Bi-dimensional Ensemble Empirical Mode Decomposition (FABEEMD), is introduced. This new technique is an improvement of Bhuiyan’s work in which pairs of positive and negative signed white noise sets are added into the signal during each iteration of FABEEMD. The introduction of white noise conjugate pairs has resolved the difficulties of mode-mixing and extrema lacking as encountered in the original framework while preserving the extracted bi-dimensional intrinsic mode function (BIMFs) noise-free. As a result, only a small ensemble is required in FABEEMD, enabling the algorithm to decompose any size economically without sacrificing the fidelity. The required smoothed topography is then constructed by reassembling the low frequency components upon the model toleration on surface roughness. The decomposition shows that the proposed method consistently performed much better than the original framework in distinguishing and extracting the local features of different surface roughness. As the noise amplitude increased, macro topographical variation is observed gradually shifting from high to low frequency components, giving the latter a more detailed depiction of the surface. This enables the smoothing to adapt to the surface roughness requirement of the flood simulation model.

The second strategy adopted in this work is to develop a simplified flood inundation model by replacing the shallow water equations with Manning’s formula to omit the nonlinear derivatives from momentum computation. The computational domain are spatially discretized into a Triangulated Irregular Network (TIN) with each element being treated as a storage tank. To spatially distribute the runoff with 1D uniform flow formula, we incorporate the Manning’s formula with a multiple flow direction (MFD) framework to form a two steps algorithm in which the total outflow of an element is first computed through weighted averaging the variables required in Manning’s formula, then partitioned into directional components according to their corresponding hydraulic gradients. For urban flooding, underground sewer system and rainwater harvesting are incorporated with the model to provide more reliable simulation. The model was applied to Linbian River watershed during Typhoon Morakot of August 2009 and an idealized urban terrain inundated by a designed rainstorm. In the former, the model has shown satisfactory results in terms of inundation extent and depth by verifying against in-situ observation and the simulation of a 2D hydrodynamic model. For urban flood modeling, the simulation is shown reasonable and stable. In both cases, 0% of mass loss was achieved.

並列關鍵字

Bi-dimensional ensemble empirical mode decomposition (BEEMD) ； Topography smoothing ； Flood inundation simulation ； Storage cell approach ； Multiple flow direction algorithm (MFD)

參考文獻

Bhuiyan, S. M., Adhami, R. R., & Khan, J. F. (2008a). A novel approach of fast and adaptive bidimensional empirical mode decomposition. IEEE International Conference on Acoustics, Speech and Signal Processing (pp. 1313-1316). Las Vegas: IEEE.

Bhuiyan, S. M., Adhami, R. R., & Khan, J. F. (2008b). Fast and adaptive bidimensional empirical mode decomposition using order-statistics filter based envelope estimation. EURASIP Journal on Advances in Signal Processing, 2008.

Braun, S., & Feldmann, M. (1997). Time-frequency characteristics of nonlinear systems. Mechanical Systems and Signal Processing, 11(4), 611-620.

Damerval, C., Meignen, S., & Perrier, V. (2005). A fast algorithm for bidimensional EMD. Signal Processing Letters, 12(10), 701-704.

Dong, W. P., & Stout, K. J. (1995). Two-dimensional fast Fourier Transform and power spectrum for surface roughness in three dimensions. Journal of Engineering Manufacture, 205(5), 381-391.