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就接近于实际介质的参考介质来说,合成地震记录的计算可线性化。如果利用最小平方法公式,则反演问题可以当作二次最优化问题来对待。如果参考介质是对称的,那么反演问题就能大大地被简化。对于均质参考介质,通过对所有空间变量的傅里叶变换,即可以得到精确而又经济的解。特别是,这种解利用不必分解任何线性系统的显示公式便能获得(不在傅里叶域内作处理时通常就是这种情况)。可是,均质参考介质的假设一般总是不切合实际的。在某些情况下,参考介质可以与深度有关。因此,很明显,除深度外,通过对时间和所有空间变量的傅里叶变换,反演问题也可获得一种精确而又经济的解。如果δm(x,z)在参考介质和实际(二维)介质之间是(未知的)不同的,那么对于每一个水平波数 k_x 的值来说,解其维数等于深度取样值的线性系统,也是能够获得傅里叶变换的解δm(k_x,z)的。
For a reference medium close to the actual medium, the calculation of synthetic seismograms can be linearized. If the least square method is used, the inversion problem can be treated as a quadratic optimization problem. If the reference medium is symmetric, then the inversion problem can be greatly simplified. For a homogeneous reference medium, an accurate and economical solution can be obtained by Fourier transformation of all spatial variables. In particular, this solution can be obtained using display formulas that do not have to be decomposed into any linear system (this is usually not the case when processing within the Fourier domain). However, the assumption of homogeneous reference media is generally not realistic at all. In some cases, the reference medium can be depth dependent. Therefore, it is clear that, with the exception of depth, an accurate and economic solution can also be obtained by the Fourier transform of time and all spatial variables, the problem of inversion. If δm (x, z) is (unknown) different between the reference medium and the actual (two-dimensional) medium, then for each value of horizontal wave number k_x, a linear system whose dimension is equal to the depth sample is solved , It is also possible to obtain the solution of the Fourier transform δm (k_x, z).