基于初至波走时层析成像的Tikhonov正则化与梯度优化算法

Tikhonov regularization and gradient descent algorithms for tomography using first-arrival seismic traveltimes

初至波走时层析成像是利用地震初至波走时和其传播的射线路径来反演地下介质速度的技术.该问题本质上是一个不适定问题,需要使用正则化方法并辅之以适当的最优化技巧.本文从数值优化的角度介绍了初至波走时层析成像的反演原理,建立了Tikhonov正则化层析成像反演模型并提出求解极小化问题的加权修正步长的梯度下降算法.该方法可以从速度模型的可行域中迭代找到一个最优解.数值试验表明,该方法是可行和有应用前景的.

With the development of seismic exploration, inversion and imaging become key issues because of complex structures. It is in urgent need to build accurate near-surface velocity models. Nowadays, three primary numerical methods are developed to acquire the velocity model, i.e., stack velocity analysis, migration velocity analysis and tomography velocity analysis. For the near-surface seismic problem, the first two methods are not suitable because of insufficient fold numbers and reflections. So the tomography method has received much more attention.First-arrival traveltime seismic tomography refers to inversion of medium velocity using first-arrival seismic wave traveltimes and their ray paths. The first task is model parameterization which discretizes the stratigraphic model into many slowness units by gridding. Secondly, based on the slowness units, the ray paths are analyzed by the shortest traveltime ray tracing. Then the traveltime equation is established to solve the velocity model. This is an ill-posed inverse problem. Proper regularization technique and optimization methods are required. Therefore, a Tikhonov regularization model with constraints on feasible set was established, and a gradient descent method with modified step sizes was also developed to obtain an optimized solution.Three different theoretical models were designed to test the new algorithm. The first is a horizontal layered model：there were three horizontal layers with the velocity of 600 m·s-1, 1200 m·s-1 and 2000 m·s-1 from top to bottom in the real model. By random disturbance of the model, we got an initial velocity model which was far away from the real model. The inversion result shows that the new algorithm can converge to the true model quickly even with poor initial condition. This shows that the new algorithm is stable and fast in convergence. Comparison with the well-known conjugate gradient （CG） method indicates that this new algorithm requires less memory and has higher convergence speed than the traditional CG algorithm. Specifically, the memory used by the CG algorithm is 2 times that of the new algorithm, and the running time of the CG algorithm is 1.82 times that of the new algorithm. The second is a graben velocity model. Again, by random disturbance to the model, we got an initial velocity model which was far away from the true model. The inversion results also show that the new algorithm can converge quickly even with poor initial conditions and obtain a satisfactory result. The third is a fault velocity model with irregular interfaces. The inversion results show that the new algorithm could converge quickly with poor initial conditions and obtain a satisfactory result even if the model is complex.Based on the theory of first-arrival traveltime seismic tomography, a Tikhonov regularization model with constraints on feasible set was established, and a gradient descent method with modified step sizes was also developed to obtain an optimized solution. The new method has three characters：（1） In forward modeling, the shortest traveltime ray tracing is suitable for complex models with highly changing velocities. Sources and receivers can be arranged arbitrarily. The running time has no correlation with the degree of complex structure of the model. And a 3D calculation is easy to perform. （2） Aiming at the ill-posed problem, a proper regularization technique was developed to make the solution converge stably. （3） This method can find an optimal solution from the feasible region of the velocity model by iterations.