本文選題：三維電磁法 + 積分方程法��； 參考：《吉林大學》2015年碩士論文

【摘要】：自上世紀20年代興起以來，電磁法在礦產(chǎn)資源勘探和地殼構造的研究中一直起著非常重要的作用。長期以來，電磁法的理論研究一直局限在一維情況，但是對于復雜的地質條件，一維理論存在極大的局限性。積分方程在三維數(shù)值模擬中存在較大優(yōu)勢，對小型異常體只需要對異常體本身進行剖分，避免了使用微分方程法時在內(nèi)存上的需求，但是積分方程方法也存在相當?shù)木窒扌裕簩τ诖笮彤惓ｓw，由于積分方程離散所得的是稠密矩陣，對計算機的內(nèi)存要求急劇增加；矩陣求逆對大型矩陣來說幾乎不可行，所以需要采用迭代法來求解，然而對于大型矩陣，迭代時需要進行的矩陣向量乘積所需的計算時間不容忽視；此外，由于格林函數(shù)的限制，傳統(tǒng)的積分方程只適合于層狀介質中相關問題的求解。這些問題都限制了積分方程技術的應用。針對這些問題，本文提出了一些改進方法。 通過利用系數(shù)矩陣的Toeplitz性質，使傳統(tǒng)積分方程方法中遇到的稠密矩陣的存儲困境得到了有效地解決。同時，，利用Toeplitz矩陣所具有的特殊性質，用快速傅里葉變換來實現(xiàn)了迭代算法中的矩陣向量乘積，加速了傳統(tǒng)積分方程技術中直接求矩陣向量乘積的過程。由于復雜的地質模型都可以轉化為多個異常體的問題，本文利用不均勻背景電導率的概念，在考慮異常體間耦合的基礎上，給出了對多個異常體進行響應模擬的方法，并將這種方法與不考慮耦合時的響應進行了對比。 通過本文的工作，得出以下結論：（1）對積分方程離散的系數(shù)矩陣按性質可分解為兩項，其中一項滿足三重Toeplitz矩陣的特征，另一項滿足Hankel-兩重Toeplitz矩陣的特征，而Hankel矩陣可以通過簡單的置換矩陣轉化為Toeplitz矩陣；（2）由于Toeplitz矩陣存在的特殊性質，在矩陣的存儲時只需要存儲各重矩陣的第一行和第一列，這樣，當剖分網(wǎng)格數(shù)量很多時，這種存儲可以節(jié)省大量內(nèi)存；（3）對Toeplitz矩陣，可以使用快速傅里葉變換實現(xiàn)矩陣與向量的乘積，這種方法可以高效準確地得到解；（4）用考慮耦合效應的方法計算多異常體的響應更適合計算相距較遠的多個異常體響應，所得到的響應與不考慮耦合時的響應結果是一致的。
[Abstract]:Electromagnetic method has been playing an important role in the exploration of mineral resources and the study of crustal structure since its rise in 1920s. For a long time, the theoretical study of electromagnetic method has been confined to one-dimensional case, but for complex geological conditions, one-dimensional theory has great limitations. The integral equation has a great advantage in 3D numerical simulation. It only needs to divide the abnormal body itself to avoid the memory requirement when using differential equation method. However, the integral equation method also has some limitations: for large abnormal bodies, because the discrete integral equations are dense matrices, the requirements for computer memory are increased dramatically, and the inverse matrix is almost impossible for large matrices. So it is necessary to use iterative method to solve the problem. However, for large matrices, the computing time required for the product of matrix vectors in iteration cannot be ignored; in addition, due to the restriction of Green's function, The traditional integral equations are only suitable for solving related problems in layered media. These problems limit the application of integral equation technique. In order to solve these problems, some improved methods are proposed in this paper. By using Toeplitz property of coefficient matrix, the storage dilemma of dense matrix in traditional integral equation method is solved effectively. At the same time, by using the special properties of Toeplitz matrix, the matrix vector product in iterative algorithm is realized by using fast Fourier transform, which accelerates the process of directly calculating matrix vector product in traditional integral equation technology. Because complex geological models can be transformed into multiple anomalous bodies, this paper presents a method to simulate the response of multiple anomalous bodies based on the concept of non-uniform background conductivity and considering the coupling between anomalous bodies. By comparing this method with the response without considering coupling, the following conclusion is drawn: 1) the discrete coefficient matrix of integral equation can be decomposed into two terms according to its properties, one of which satisfies the characteristic of triple Toeplitz matrix. The other satisfies the characteristic of Hankel-double Toeplitz matrix, and Hankel matrix can be transformed into Toeplitz matrix by simple permutation matrix.) because of the special property of Toeplitz matrix, only the first row and the first column of each multiple matrix need to be stored in the storage of the matrix. In this way, when the number of meshes is large, this storage can save a lot of memory. The Toeplitz matrix can be used to realize the product of matrix and vector by using fast Fourier transform. This method can be used to calculate the response of multiple anomalous bodies efficiently and accurately. The response of multiple anomalous bodies is more suitable to be calculated by considering the coupling effect. The results obtained are in agreement with those obtained without considering the coupling.
【學位授予單位】：吉林大學
【學位級別】：碩士
【學位授予年份】：2015
【分類號】：P631.325

【引證文獻】

相關博士學位論文 前1條

1 賁放;海洋可控源電磁法三維正反演理論與各向異性影響機制研究[D];吉林大學;2016年

相關碩士學位論文 前2條

1 鄭建波;基于小波方法的CSAMT靜態(tài)校正[D];吉林大學;2016年

2 朱成;帶地形頻率域可控源電磁法三維正反演研究[D];吉林大學;2016年

本文編號：2005471