一维声波方程声阻抗反演问题的稳定性分析

THE STABILITY ANALYSIS FOR THE INVERSE PROBLEM OF ONE DIMENSIONAL ACOUSTICS WAVE EQUATIONS

讨论了一维声波方程的声阻抗反演问题。对于脉冲激发的情况，得到反问题适定的充分、必要条件，适定条件的形式非常简单，在实际计算中易于判定。对于子波激发的情况，由子波响应反演声阻抗通常是不适定的，不适定程度与子波的奇性有关，子波越光滑，相应的反问题的不适定程度越高。同时给出了定量的说明，为一维声波方程声阻抗反演问题的数值计算提供了理论。

The inverse problem for the one dimensional acoustic wave equations is discussed.For the inverse problem to determine acoustic impedance from impulse response,sufficient condition and necessary condition are obtained for the wellposedness.These conditions are very simple and can be verified easily in numerical computation.For the inverse problem to determine acoustic impedance from wavelet response,it's usually illposed.The detailed analysis indicates that the more smooth the wavelet is,the more illposed the inverse problem is.