基于Chebyshev自褶积组合窗的有限差分算子优化方法

Optimized finite-difference operators based on Chebyshev auto-convolution combined window function

有限差分法广泛应用于地震波数值模拟、成像和波形反演中,差分数值解的精度直接影响着地震成像和反演的效果.因为有限差分算子可以通过截断伪谱法的空间褶积序列得到,而截断窗函数的属性影响有限差分算子逼近微分算子的精度.具体地讲,窗函数的幅值响应的主瓣和旁瓣决定了有限差分算子逼近的精度,主瓣越窄,旁瓣衰减越大,则有限差分算子逼近微分算子的精度越高,更好地压制数值频散.基于此认识,本文提出了一种基于Chebyshev自褶积组合窗截断逼近的有限差分算子优化方法.Chebyshev自褶积组合窗的主瓣较窄,且旁瓣衰减大,其可通过只调节三个参数,更直观和可视化地控制主瓣和旁瓣的形状,改变有限差分算子逼近微分算子的精度;该窗函数截断逼近的有限差分算子不仅有较大的谱覆盖范围,而且精度误差波动较小,这表明低阶的差分算子可以达到高阶算子的精度,且逼近误差更稳定;从经济上来讲,将有效地减少模拟计算花费,提高计算效率.

The finite-difference method has been widely utilized in seismic wave numerical modeling,seismic imaging and full waveform inversion.The accuracy of finite-difference numerical solutions directly affects results of seismic imaging and inversion.Using an autoconvolution combined window function to truncate spatial convolutional counterpart of the pseudospectral method,optimized explicit finite-difference operators are derived. The truncated window function method is used to get optimized finite-difference operators.Firstly,we analyze the influence on the accuracy of finite-difference operators caused by the properties of main lobe and side lobe in the amplitude response of truncated window functions.Secondly,based on the methods of auto-convolution and weighted combination,a window function which has narrower main lobe and larger attenuation of side lobe is designed,correspondingly bringing higher accuracy of finite-difference approximation.Finally,we use the window function to get optimized finite-difference operators. From the analysis of window functions,the factors that affect the accuracy of finitedifference approximation can be summarized in two aspects：（1）The width of main lobe of window functions.（2）The attenuation of side lobe of window functions.The window functions which have narrower main lobe and larger attenuation of side lobe can yield higher accuracy of finite-difference approximation.The Chebyshev window function has appropriate width of main lobe and attenuation of side lobe to get better stability of accuracy error on the premise of maintaining the appropriate wave number coverage range.Furthermore,the auto-convolution method will increase the attenuation of side lobe,however,widen the main lobe.Weighted combination can remedy the defect,choose different weight coefficients to reduce the width of main lobe.Combining the two methods,a specific window function named Chebyshev autoconvolution combined window is designed.Compared with the accuracy curves of approximation between the finite-difference operators truncated by Chebyshev auto-convolution combined window and the conventional operators,the former lead to great accuracy in a bigger frequency region.Tests on a homogeneous model and the Marmousi model show that the dispersion caused by the operators based on the Chebyshev auto-convolution combined window is quite weak under the same order of the conventional operators.Further comparison with the finite-difference operators truncated by improved binomial window,our operators still have a bigger frequency coverage range and smaller fluctuation of accuracy error. The finite-different operators based on the Chebyshev auto-convolution combined window have higher accuracy than that of the conventional operators and operators based on the improved binomial window.Our optimized eighth-order and twelfth-order operators can respectively reach,even exceed the accuracy of the conventional twelfth-order and twenty-fourth-order finitedifference operators,and the maximum deviation of absolute error is within [0,0.0004].For higher-order operators,the accuracy increase becomes more obvious.The results of elastic wavefield numerical modeling demonstrate that the operators based on the Chebyshev autoconvolution combined window can efficiently suppress the numerical dispersion and has greater modeling accuracy under the same discretizations without extra computing costs.In addition,through adjusting parameters of the auto-convolution combined window,we can adjust the accuracy of finite-difference approximation as required,visually and intuitively.