應用小波轉換及消噪於線性擾動響應模式之研究

Linear Perturbation Response Model Based on Wavelet Transform and Denoise

周建明(Chien-Ming Chou) ； 鄭士仁(Shin-Jen Cheng)

《農業工程學報》 57卷1期 (2011/03) Pp. 1-15

https://doi.org/10.29974/JTAE.201103.0001

摘要

本研究將傅統線性擾動響應摸式用於求得平滑季節平均值之傅立葉級數，改採小波轉換及小波消噪二種方法分別進行處理。首先對降雨及逕流時間序列分別進行M層小波分解，可獲致在M個尺度下之光滑近似訊號及細節訊號。第一種應用方法係將第M個尺度之光滑近似訊號視爲平滑季節平均值，針對M層之細節訊號使用M個線性擾動響應模式分別模擬之，即假設每一層輸入與輸出之細節訊號間爲線性關係，最後透過逕流之平滑季節平均值與M個擾動項推估值之小波重構可獲致輸出推估值。第二種應用方法係令各分辨層小於某門檻值之細節訊號爲零，並將各分辨層經此處理後之細節訊號及第M層之光滑近似訊號，利用小波重構以獲致消噪後之降雨及逕流水文時間序列，並將其視爲平滑季節平均值，而將原始時間序列减去消噪後時間序列所得到之噪音俯視爲擾動項。假設輸入噪音與輸出噪音間之關係爲線性關係，採用最小二乘法推估此單一響應函數後，再利用降雨擾動項計算相對應之逕流擾動項推估值，最後將消噪後之逕流加上逕流擾動推估值，可獲致逕流量輸出推估值。上述二種方法經應用於基隆河流域五堵上游集水區之日降雨－逕流歷程分析，研究結果顯示應用小波轉換及小波消噪，均可以有效地計算平滑之季節平均值及相對應之擾動項，進而提高降雨－逕流歷程模擬之精確度。

關鍵字

小波轉換 ； 小波消噪 ； 線性擾動響應模式 ； 降雨－逕流歷程

並列摘要

This work models rainfall-runoff processes using a novel framework based on wavelet analysis and the linear perturbation response model (LPRM). Wavelet transforms are substituted for the Fourier series to acquire smooth seasonal means and perturbation terms from rainfall-runoff data in the LPRM. The observed rainfall and runoff time series are decomposed using wavelet transforms to obtain the approximation and detailed signals of rainfall and runoff, respectively, at M resolution levels. In the first application, the detailed signal at each resolution level is considered the perturbation term utilized in the LPRM. We assume the relationship between the input detailed signal and output detailed signal at each resolution level is linear. Moreover, the approximation at resolution level M corresponds to the smooth seasonal mean employed in the LPRM. The estimated runoff can be derived from the approximation of runoff at resolution level M and estimated detailed signals of runoff at all resolution levels using wavelet reconstruction. In the second application, the values of a detailed signal below a certain threshold are set to zero at each resolution level, meaning denoise. The denoisy rainfall and runoff time series can be obtained from the approximation and denoisy detailed signals at all resolution levels of rainfall and runoff, respectively, using wavelet reconstruction. The denoisy rainfall and runoff time series are then regarded as the smooth seasonal mean employed in the LPRM. The noise, i.e., original time series minus denoisy time series, is considered the perturbation term employed in the LPRM. Moreover, we assume the relationship between input noise and output noise is linear. The summation of the denoisy runoff and estimated noise of runoff yields the overall estimated runoff in the LPRM. To verify the accuracy of the proposed model, this work chose daily rainfall-runoff data for upstream of the Kee-Lung River as a case study. Analytical results demonstrate that wavelet transform and denoise can accurately estimate the smooth seasonal mean and perturbation terms, thereby enhancing the accuracy of modeling of rainfall-runoff processes.

並列關鍵字

Wavelet transform ； Wavelet denoise ； Linear perturbation response model ； Rainfall-runoff processes