A comprehensive solution to the linear deconvolution problem
A comprehensive solution to the linear deconvolution problem
复制标题
线性反卷积问题的综合解决方案
DOI：
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发表时间：
1981
期刊：
影响因子：
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通讯作者：
D. Oldenburg
中科院分区：
文献类型：
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作者：
D. Oldenburg
总结
当模糊函数或源小波（大约）已知时，线性逆理论用于对数据集进行反卷积。本文没有尝试从无数个模型中找到一个适合数据的模型，而是使用巴科斯和吉尔伯特的方法来生成模型的局部平均值，该平均值除了数据误差引起的统计不确定性之外是唯一的。平均值、统计误差和相关的平均窗口完全整理了我们关于模型的知识。可以通过牺牲分辨率来获得具有较低标准偏差的平均值，并且研究者可以自由选择那些最有意义的结果。此外，该方法是最佳的，因为无法构建具有更大分辨能力但仍产生具有相同统计精度的平均值的其他平均窗口。
我们的时域反卷积方法与寻找源小波的逆滤波器非常相似，实际上平均窗口只是这两个函数的卷积。然而，通过研究分辨率和精度之间的权衡，我们发现数据误差可能比维纳最佳逆滤波器的参数（例如滤波器的长度和输出尖峰的所需位置）重要得多。
在频域中，已经开发了用于权衡精度和分辨率的方程，并且计算特别简单，因为不需要矩阵求逆。将提供足够的例子来表明将观测误差纳入反卷积过程的重要性。此外，我们还展示了如何通过将平均窗口整形为预定宽度的高斯函数来减少平均窗口的旁瓣，研究了使用地震问题的零面积源函数特征的效果，并尝试在小波仅近似已知时进行反褶积。这里导出的频域反卷积滤波器也与 Helmberger、Wiggins 和 Deregoiwski 直观导出的频域反卷积滤波器进行了定量比较。
最后，我们展示了如何使用平均值和平均窗口中的信息来构建参数模型，该模型由一系列拟合数据的 delta 函数组成。这种模型在地震学和光谱研究中具有重要意义。
Summary
Linear inverse theory is used to deconvolve a data set when the blurring function or source wavelet is (approximately) known. Rather than attempting to find one of infinitely many models which fits the data this paper uses the methods of Backus and Gilbert to generate localized averages of the model which are unique except for a statistical uncertainty caused by errors in the data. The averages, their statistical error and the associated averaging window completely codify our knowledge about the model. Averages with lower standard deviations can be had by sacrificing resolution and the investigator is free to choose those results which are most meaningful. Moreover, this method is optimum in the sense that no other averaging window can be constructed which has greater resolving power and yet produces averages with the same statistical accuracy.
Our deconvolution method in the time domain is shown to be very similar to finding the inverse filter of the source wavelet, and indeed the averaging window is simply the convolution of these two functions. However, by investigating the trade-off between resolution and accuracy we have shown that the data errors can be much more important than the parameters of the Wiener optimum inverse filter such as the length of that filter and the desired location of the output spike.
In the frequency domain the equations for trading off accuracy and resolution have been developed and the computations are seen particularly to be simple because no matrix inversion is required. Sufficient examples will be presented to show the importance of incorporating the observational errors into the deconvolution procedure. Additionally, we have shown how to reduce the sidelobes of the averaging windows by shaping them into Gaussian functions of a predetermined width, have looked at the effects of using a zero area source function characteristic of seismic problems, and have attempted a deconvolution when the wavelet was only approximately known. The frequency domain deconvolution filter derived here is also compared quantitatively with those derived intuitively by Helmberger and Wiggins and Deregoiwski.
Lastly, we show how information in the averages and averaging windows can be used to construct a parametric model, composed of a series of delta functions, which fits the data. Such a model is of importance in seismological and spectroscopic studies.