Mini-batch Stochastic Gradient Descent Method Applied To Seismic Tomography

Full waveform inversion that is an inversion method based on wave equation,have been popular in recent years.It is more rigorous and has higher resolution than ray travel-time tomography based on high frequency ray approximation.The numerical methods for solving wave equation are such as finite difference method,finite element method and spectral element method,which usually needs a great amount of computa-tion time and disk space,especially for three-dimension inversion problem.It’s neces-sary to compute the sensitivity kernel,and at least two forward simulations are needed for each seismic source,one for the forward wavefield,the other for adjoint wavefield.Then,the computation of kernel function when calculating gradients needs two forward simulations for all sources,which becomes a very apparent problem when the number of sources is large.To solve the problem,people prefer to parallel computing,such as dividing com-putation regions into many MPI blocks;besides the parallelism of computation regions,sources can be done the same,which however will lead to the problem of high RAM requirements.In addition to the parallelism in CPU architecture,the GPU parallel tech-nology is also very popular in recent years that also encounter RAM problems.There-fore,when the computing resources are limited,we need to seek an inversion method that is fairly cheap and efficient.The mini-batch stochastic gradient descent method proposed in this paper,which is popular in machine learning and very useful in deal-ing with big data and online learning.For the full waveform inversion,it needs only two forward simulations for each pair of one source and many receivers,and it’s na-ture to make it as a mini batch.In contrast to those traditional methods,each update only requires the forward and adjoint wavefield simulations for each source,then the sensitivity kernel is computed and the model is updated immediately.In this paper,we will apply the mini-batch SGD method onto the ray traveltime inversion and full waveform inversion.The formulas of sensitivity kernel are given for both the traveltime tomography based on ray theory and the waveform inversion based on finite-frequency theory in chapter 2.In chapter 3,some numerical examples are tested,and the mini-batch SGD method will be compared with other traditional gradient-based inversion methods,how to choose suitable parameters also talked.Besides,the applications of mini-batch SGD method onto 4D inversions are studied.How to choose a suitable misfit function is also very important issue for the full waveform inversion problem.The common strategy is that the inversion is running from low frequency to high frequency in full waveform inversion,through a low-pass filter applied onto data.In this paper,we adopt another strategy based on wavelet trans-form,by defining the time-frequency misfit function and choosing proper weighting functions.