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Viscoelastic wave finite difference modeling in the presence of topography with adaptive free surface boundary condition

Dong Shu-Li, Chen Jing-Bo, Li Zheng

Acta Geophysica

2021

Vol. 69, no. 6

2205--2217

An accurate free-surface boundary condition is significant for seismic forward modeling and inversion. The finite-difference method (FDM) is widely used for its simplicity and efficiency. However, unlike the finite-element method (FEM) satisfying naturally the stress-free condition at the free surface, FDM needs additional treatment, particularly in the presence of irregular topography. In the elastic wave finite-difference modeling, the adaptive parameter-modified free-surface boundary condition has the advantages of high accuracy and simple operation. The viscoelastic wave equation can describe the nature of seismic waves more realistically. Based on the staggered-grid FDM, we extend the adaptive free-surface boundary condition to the viscoelastic medium with topography. This approach involves a combination of the average medium theory, vacuum approximation and limit idea. It is realized by modifying the viscoelastic constitutive relation. This method is simple enough, because three types of grid elements and in fact only two kinds of expressions are enough in the presence of topography. We only need to deal with the Lamé parameters and the density at the free surface without reconstructing the existing algorithm. Viscoelastic analysis of different quality factor settings shows the viscous effect. Numerical examples display that the results of the presented method agree well with the reference solutions of spectral-element method both in crest- and trough-like model and in simplified Foothill model with irregular topography. The simulation of original Foothill model demonstrates the feasibility of our method.

Frequency domain elastic wave modeling for polygonal topography using rotated average derivative diference operators

Li Zheng, Chen Jing-Bo, Cao Jian

Acta Geophysica

2020

Vol. 68, no. 5

1387--1409

Modeling of seismic wave propagation in areas with irregular topography is an important topic in the feld of seismic exploration. As a popular numerical method for seismic modeling, the fnite diference method is nontrivial to consider the irregular free surface. There have been extensive studies on the time-domain fnite diference simulations with irregular topography; however, the frequency-domain fnite diference simulation considering irregular topography is relatively less studied. The average-derivative approach is an optimal numerical simulation scheme in the frequency domain, which can produce accurate modeling results at a relatively low computational cost. Nevertheless, this approach can only deal with the modeling problems with a fat free surface. To address this issue, we design a new frequency-domain fnite diference scheme by introducing the polygonal representation of topography into the average-derivative method. The irregular topog raphy is represented by line segments with various slopes. An extension of the conventional average-derivative diference operator in the local rotated coordinate system is used for formulating the spatial derivatives aligned with the topographic line segments. As a result, new average-derivative diference schemes are obtained for irregular topography. In this way, the average-derivative optimal method is generalized to the model with irregular topography. Numerical examples show the efectiveness of the presented method.