2.1 Very Fast Simulated Annealing Method
Since it was proposed by Kirkpatrick et al.［15］in 1983，the simulated annealing algorithm have become an important calculation method for nonlinear optimum problem with the advantage of obtaining global optimum solution.The conventional simulated annealing algorithm includes the following steps.（1）Select an initial model m0randomly in the range of variation of the model parameters，and compute the cost function E（m0）.（2）Produce a new model m by the perturbation of the current model m0，and then compute the cost function E（m0）and the discrepancyΔE=E（m）-E（m0）.（3）IfΔE＜0，the new model m is accepted；ifΔE＞0，the new model m is accepted by the dependence on probability P=exp（-ΔE/T），where T is the temperature.If the new model m is accepted，then m 0=m，E（m0）=E（m）（4）At temperature T，the process of perturbation and acceptance is repeated certain times，and it means step（2）and step（3）are repeated.（5）Decrease the temperature T slowly.（6）Repeat step（2）and step（5）until convergence condition is satisfied.The conventional simulated annealing algorithm consists of two parts.One is to make the random perturbation of the model and to calculate of the change of the energy/cost function.The other is to accept the model in accordance with probability P.When the temperature is high，bad models which increase the cost function can also be accepted，and then the method can jump off local minimum.When the temperature is low，only good models can be accepted.Global optimum solution can be obtained by the method when the temperature decreases slowly from enough high level.
In the very fast simulated annealing algorithm proposed by Ingber［16］，the new model is produced by quasi-Cauchy distribution which depends on temperature.The expression is
mi=m0i+yi（Bi-Ai），(https://www.daowen.com)
yi=Tsign（u-0.5）［（1+1/T）|2u-1|-1］，
where m 0i is the ith variable of current model.u is a random number homogeneously distributed in the range of 0～1，［Ai，Bi］denotes the range of miand the calculated mimust satisfy mi∈［Ai，Bi］.sign（）denotes sign function.This improvement increases the convergence speed of simulated annealing algorithm.In our inversion，interval velocities are model parameters and the average-square-difference between the calculation and the observation of the travel-time in theτ-p domain is defined as the cost function.
