基于一种改进凸集投影方法的地震数据同时插值和去噪

An improved projection onto convex sets method for simultaneous interpolation and denoising

基于稀疏反演的地震插值方法是一种重要的插值方法,然而大多数这类方法只针对无噪声数据或者高信噪比数据插值.实际上,地震数据含有各种噪声,使得插值问题变得更加困难.凸集投影方法是一种高效的插值算法,但是对于含噪声数据的插值效果不理想,针对含噪声数据提出的加权凸集投影方法能够实现同时插值和去噪,但是除了最小阈值需要认真选取外,增加一个权重因子来实现去噪功能.本文由迭代阈值算法推导出加权凸集投影方法,证明其是解无约束优化问题的一种方法,加权因子可以看作拟合误差项的系数.本文还提出了一种改进的凸集投影方法,与原始凸集投影方法相比该方法不需要增加任何计算量,只要通过阈值的选择来进行插值和去噪.数值模拟证明了该算法的计算效率,并且对含噪声数据能够实现较好的插值效果;先插值后去噪的结果证明了同时去噪和插值算法的可靠性和稳定性.

Seismic interpolation can provide a complete wavefield from incomplete wavefield information and is crucial for some processings such as multiple elimination,wave equation migration and denoising.Currently,most interpolation methods are suitable for high signal-tonoise ratio data.However,filed seismic data often contain different kinds of noise,especially the random noise.The existence of noise can deteriorate the result of interpolation.The weighted projection onto convex sets（POCS）method is a method for random noise contained data interpolation.However,there is no specific explanation of this method except the iteration formula.This paper analyzed the algorithm theory of the weighted POCS method,and provedthat it is a method for unconstrained optimization problem,the weighting factor can be regarded as the coefficient of the fitting error.Meanwhile,we also proposed an improved POCS method for simultaneous interpolation and denoising.This algorithm requires no additional computational effort compared with the original POCS method.Seismic interpolation especially the high dimensional interpolation belongs to large-scale computational problems,thus fast algorithms should be chosen to improve the efficiency of seismic interpolation.Sparse transform methods change the interpolation problem into a sparse optimization based on the assumption that seismic data can be expressed sparsely in some transform domain.POCS method is such a method with high efficiency and robust results.However,this method can not get acceptable results for random noise contained data interpolation.The weighted POCS method is a method for random noise contained data interpolation on the basis of the POCS method.But there is no specific explanation of this method up to now.This paper deduced the weighted POCS method from the iterative thresholding method,we proved that the weighted POCS method is a method for unconstrained optimization,the weighted factor can be seen as the coefficient of the fitting error.However,the original POCS method is aiming at an equality constrained optimization.This is the essential difference between them. Meanwhile, we also proposed an improved POCS method with the same computational complexity as the original POCS method.The improved POCS method can realize interpolation and denoising simultaneously by adjustment of the least thresholds.In order to test the interpolation and denoising capability of the improved POCS method,four data sets were used to make numerical experiments.The first experiment is a synthetic noiseless data interpolation.The dimension of the synthetic data is 256×256with trace distance25 mand time sampling rate 4ms.The original POCS method is chosen as the benchmark.Both the computational efficiency and numerical result are comparable for them.The second data is field data with 115 traces.The trace distance is 25 mand the time sampling rate is 4 ms.This experiment also proved that both of them can get comparable results for noiseless data interpolation.The third example is a noise contained data interpolation.The original POCS method can not get acceptable result no matter how to choose the least thresholds,while the improved POCS method can provide good result by choosing the least thresholds carefully.The fourth example also proved that the original POCS method failed in the noise contained field data interpolation,and the improved POCS method is suitable for noise contained data interpolation.Numerical experiment on first-interpolation-then-denoising proved the rationality and robustness of simultaneous interpolation and denoising.We deduced the weighted POCS method from the iterative thresholding method,and proved that it is a method for unconstrained optimization,the weighting factor is the coefficient of the fitting error. However,the original POCS method is a method for equality constrained optimization.This is the essential difference between them.We also proposed an improved POCS method to realize simultaneous interpolation and denoising.This method just need to adjust the least thresholds to realize noiseless data and random noise contained data interpolation.The computational effort of the improved POCS method is the same as the original POCS method but with more broad applications.