波动方程保辛近似解析离散化算法研究
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摘要
正演数值模拟算法是反演的重要基础，而差分方法又是目前地震勘探领域应用最为广泛的正演手段之一。本文提出了一类新的求解地震波方程的正演差分方法。首先，通过在传统的波动方程哈密尔顿系统中引入位移与粒子速度的空间梯度，建立了波动方程扩充的哈密尔顿系统。然后，针对波动方程扩充的哈氏系统，发展了一类具有保辛和低数值频散特性的数值方法，称为近似解析保辛分部龙格库塔（NSPRK）方法。其后，针对NSPRK方法进行了详细的理论分析，包括其误差、稳定性条件、数值频散关系等。理论分析表明，与传统辛算法相比，尽管NSPRK类方法稳定性条件略严格，但数值频散误差较小。
为了进一步考察本文所提出新方法的有效性和优越性，论文采用NSPRK方法计算得到了三维声波与弹性波方程的数值解，并与解析解进行了比较，二者能够很好地吻合，从而证实了NSPRK方法的正确性。此外，本文还就NSPRK的计算效率和存储需求进行了分析和比较，结果表明，在同样消除数值频散的条件下，NSPRK(2,4)方法计算效率最高。例如，与传统的保辛格式SPRK(4,4)相比，NSPRK(2,4)的计算速度提高了约10.4倍，内存需求和并行通信量仅有其14.8%和7.6%。
另一方面，为了解决波场模拟中的人工边界问题，本文提出了一种将目前流行的分裂波场完全匹配层(PML)吸收边界条件与NSPRK相结合的新方法，有效地消减了由人工边界引起的反射波。另外，本研究还推导了二阶地震波方程所对应的卷积PML条件。数值实验表明，与分裂波场PML相比，卷积PML的内存需求减少了32%，计算速度提高了50倍以上。
最后，本文使用NSPRK(2,4)方法和两类PML吸收边界条件，给出了许多波场模拟结果，进一步证实了NSPRK方法的正确性、低数值频散特性和与PML条件结合的有效性，并研究了地震波在不同地质模型和复杂地质条件下的传播规律，获得了不少有意义的波传播新认识。
Forward modeling method is an important basis of inversion problems. Currentlyin seismic exploration, one of the most widely applied forward methods is finitedifference. This dissertation proposes a new type of finite difference method for solvingseismic wave equations. At first, a new extended Hamiltonian system for waveequations is established by introducing the spatial gradients of the displacement andparticle velocity into the phase space of the Hamiltonian system. Subsequently, forsolving the extended Hamiltonian system, a new type of symplectic numerical methodwith low numerical dispersion is developed, which is called the nearly-analyticsymplectic partitioned Runge-Kutta (NSPRK) method. Afterwards, the theoreticalproperties of the NSPRK method are analyzed in detail, including the theoretical error,stability condition, and numerical dispersion relation. The results show that, comparedwith conventional symplectic method, although the NSPRK method has a relativelytighter stability condition, it can suppress numerical dispersion effectively.
To verify the validity of NSPRK, numerical experiments for3D acoustic wave andelastic wave equations are made, which show that the numerical solutions generated byNSPRK are well coincident with the analytic solutions. Under the condition that visiblenumerical dispersions are eliminated, the computational efficiencies of NSPRK and fourconventional schemes are compared. The result shows that the NSPRK(2,4) is the mostefficient of them. For example, compared with the conventional symplectic methodSPRK(4,4), the computational speed of NSPRK(2,4) is increased by10.4times, whileits memory requirement and communication time between nodes in parallel are reducedto14.8%and7.6%, respectively.
On the other hand, to resolve the artificial boundary problem in wave-fieldsimulation, the prevalent split-field perfectly matched layer (PML) absorbing boundarycondition is combined with the NSPRK in a new strategy. This strategy turns outeffective in reducing the reflected waves caused by the truncated boundaries. In addition,the convolutional PML for the second order seismic wave equation is derived. Thenumerical experiment shows that compared with the split-field PML, the convolutionalPML needs32%less memory storage, while its computational speed can be increasedby more than50times.
Finally, NSPRK(2,4) and the two kinds of PML conditions are applied towave-field simulations in both2D and3D complex geological models. The resultsfurther prove the validity of NSPRK and its property of low numerical dispersion, andmoreover confirm the effectiveness of its combination with PML conditions. On theother hand, seismic wave propagation in different types of geological models withcomplex geological conditions is studied through these numerical simulations, and somenew recognization on wave propagation is obtained.
为了进一步考察本文所提出新方法的有效性和优越性，论文采用NSPRK方法计算得到了三维声波与弹性波方程的数值解，并与解析解进行了比较，二者能够很好地吻合，从而证实了NSPRK方法的正确性。此外，本文还就NSPRK的计算效率和存储需求进行了分析和比较，结果表明，在同样消除数值频散的条件下，NSPRK(2,4)方法计算效率最高。例如，与传统的保辛格式SPRK(4,4)相比，NSPRK(2,4)的计算速度提高了约10.4倍，内存需求和并行通信量仅有其14.8%和7.6%。
另一方面，为了解决波场模拟中的人工边界问题，本文提出了一种将目前流行的分裂波场完全匹配层(PML)吸收边界条件与NSPRK相结合的新方法，有效地消减了由人工边界引起的反射波。另外，本研究还推导了二阶地震波方程所对应的卷积PML条件。数值实验表明，与分裂波场PML相比，卷积PML的内存需求减少了32%，计算速度提高了50倍以上。
最后，本文使用NSPRK(2,4)方法和两类PML吸收边界条件，给出了许多波场模拟结果，进一步证实了NSPRK方法的正确性、低数值频散特性和与PML条件结合的有效性，并研究了地震波在不同地质模型和复杂地质条件下的传播规律，获得了不少有意义的波传播新认识。
Forward modeling method is an important basis of inversion problems. Currentlyin seismic exploration, one of the most widely applied forward methods is finitedifference. This dissertation proposes a new type of finite difference method for solvingseismic wave equations. At first, a new extended Hamiltonian system for waveequations is established by introducing the spatial gradients of the displacement andparticle velocity into the phase space of the Hamiltonian system. Subsequently, forsolving the extended Hamiltonian system, a new type of symplectic numerical methodwith low numerical dispersion is developed, which is called the nearly-analyticsymplectic partitioned Runge-Kutta (NSPRK) method. Afterwards, the theoreticalproperties of the NSPRK method are analyzed in detail, including the theoretical error,stability condition, and numerical dispersion relation. The results show that, comparedwith conventional symplectic method, although the NSPRK method has a relativelytighter stability condition, it can suppress numerical dispersion effectively.
To verify the validity of NSPRK, numerical experiments for3D acoustic wave andelastic wave equations are made, which show that the numerical solutions generated byNSPRK are well coincident with the analytic solutions. Under the condition that visiblenumerical dispersions are eliminated, the computational efficiencies of NSPRK and fourconventional schemes are compared. The result shows that the NSPRK(2,4) is the mostefficient of them. For example, compared with the conventional symplectic methodSPRK(4,4), the computational speed of NSPRK(2,4) is increased by10.4times, whileits memory requirement and communication time between nodes in parallel are reducedto14.8%and7.6%, respectively.
On the other hand, to resolve the artificial boundary problem in wave-fieldsimulation, the prevalent split-field perfectly matched layer (PML) absorbing boundarycondition is combined with the NSPRK in a new strategy. This strategy turns outeffective in reducing the reflected waves caused by the truncated boundaries. In addition,the convolutional PML for the second order seismic wave equation is derived. Thenumerical experiment shows that compared with the split-field PML, the convolutionalPML needs32%less memory storage, while its computational speed can be increasedby more than50times.
Finally, NSPRK(2,4) and the two kinds of PML conditions are applied towave-field simulations in both2D and3D complex geological models. The resultsfurther prove the validity of NSPRK and its property of low numerical dispersion, andmoreover confirm the effectiveness of its combination with PML conditions. On theother hand, seismic wave propagation in different types of geological models withcomplex geological conditions is studied through these numerical simulations, and somenew recognization on wave propagation is obtained.
引文
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