摘要
将复傅氏系数同Sinc函数褶积,然后乘一个相位移因子即可获得傅氏频率域中的精确插值公式。忽略此相移因子将会在偏移剖面中产生许多假同相轴。严格说来,每一个点的插值都是所有复系数的加权和。为了加速这种褶积运算,通常只截取加权和式中的少数几项,并使所用权系数系列的两端逐渐衰减（Tapering）,以减弱因丢掉项而引起的误差。本文提供两种改进算法以增加截取权系数的精度。在第一个方法中,按照Lanczos所用的方式,对傅氏系数做线性变换,再用新系数插值。这种新的插值权系数按1/n2的规律衰减（原来的按1/n衰减,n为权系数至插值点的样值数）,从而减少截取的精度损失。在第二种改进算法中,傅氏系数同平方Sinc函数褶积,然后乘以相移因子。该算法在对数据作偏移的同时还使输入数据线性递增。同样,权系数按1/n2衰减,因而截取少数项仍有足够精度。本文以实例说明,在空间域插值可消除折返效应（Wraparound effect）。其效果与补零的作用相同,但是却不需要占用额外的计算机内存。实际上,我们还发现仅截取两项插值系数,仍然能得到很好的偏移效果,但在消除折返效应时,至少得取四项。本文还提供了一个64×64×64的三维F-K偏移的例子（输入为单脉冲）以进一步检验所用的插值方法。
Abstract
An exact fomula for interpolation in the Fourier transform domain is given by a convolution of the complex Fourier coefficients with a sine function multiplied by a phase shift factor. Omission of the phase shift factor causes false events to appear in the migrated results.To speed-up, the calculation of the coefficients for each interpolated value, it is customary to truncate the weighted sum to a few terms and cover up the error caused by omitting terms by tapering the weights that are used. Two improvements in this procedure are demonstrated which increase the accuracy for truncation to a given number of terms. In the first, following a scheme introduced by Lanczos, the coefficients are linearly transformed into new coefficients which are then used to do the interpolation. The new interpolation weights decrease like 1/n2 (the originnal weights decrease like 1/n) where n is the number of samples from the weight to the point of interpolation, and therefore can be truncated with less loss of accuracy. In the second improvement, the coefficients are convolved with a sine-squared function multiplied by a phase shift factor. This operation migrates the data and at the same time applies a linear ramp to the input data. Again, the weights decrease like 1/n2 and truncation to a few terms is accurate.It is also demonstrated that wraparound effects can be removed using interpolation in the space direction. This has the same effect as if zeros had been padded, but additional CPU memory is not used.In prattice, it was found that truncation of the interpolation series to only two terms produced good results for migration, but inclusion of at least four terms in the interpolation formula was necessary to remove wraparound events.An example of a 64 × 64 × 64 three-dimensional F-K migration is given for an impulse test section.
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