Seismic full waveform inversion FWI is one of the most attractive seismic imaging tools for estimating subsurface fluidand rock properties by taking most if not all types of seismic waves into account

based local search step to VFSA to form a hybrid optimization method for FWI. The proposed method is able to generate an accurate background velocity model, and the result is not sensitive to the choice of the starting model. The output of the hybrid optimization method can then be utilized as a starting model for local optimization FWI methods to further refine the model. METHODS In this section, we describe the proposed hybrid optimization method for FWI. The method consists of two steps: the global optimization step using VFSA, and the local optimization step using a derivative-based optimization method. Global Optimization with VFSA VFSA is a stochastic global optimization method that utilizes the concept of simulated annealing (SA) (Kirkpatrick et al., 1983). SA is a probabilistic algorithm that aims to find the global minimum of a given function by mimicking the physical process of annealing in metallurgy. The algorithm starts with a high temperature, and the temperature is gradually decreased over a certain number of iterations. At each iteration, the algorithm randomly selects a new trial solution and decides whether to accept it based on a probability function that depends on the current temperature and the difference between the current and the new solution’s objective function value. The probability function is designed to balance the exploration of the parameter space and the exploitation of the current solution. As the temperature decreases, the probability of accepting a worse solution is reduced, and the algorithm converges towards the global minimum of the objective function. VFSA extends the SA algorithm by introducing a sparse parameterization of the search space. Instead of searching for a solution in the entire parameter space, VFSA performs the search only in a small subset of the parameters, which are chosen randomly at each iteration. This reduces the computational cost and accelerates the convergence of the algorithm. The sparse parameterization is achieved by dividing the parameter space into disjoint subsets, and only one subset is selected at each iteration. The subsets are selected randomly with a probability proportional to their size. Local Optimization with Derivative-Based Method After the global optimization step with VFSA, the output of the algorithm is a set of candidate solutions that are distributed across the parameter space. Each candidate solution is a potential starting model for the local optimization step. In this step, a derivative-based optimization method, such as the quasi-Newton method, can be employed to refine the model. In the quasi-Newton method, the derivatives of the objective function with respect to the model parameters are approximated using the information from the previous iterations. The approximation is updated at each iteration using the new gradient information. The method iteratively updates the model parameters until the objective function converges to a minimum. The method is efficient for high-dimensional optimization problems and is less prone to the local minima issue than the local optimization methods that only use the objective function values. Hybrid Optimization Method for FWI The proposed hybrid optimization method for FWI consists of two steps: the global optimization step with VFSA and the local optimization step with the quasi-Newton method. The algorithm is summarized as follows:

1. Initialize the algorithm with a starting model.
2. Perform the global optimization step with VFSA to generate a set of candidate solutions.
3. For each candidate solution, perform the local optimization step with the quasi-Newton method to refine the model.
4. Select the best solution from the refined models as the final output. The starting model in step 1 can be chosen based on prior information or a simple model derived from low-resolution seismic data. The final output in step 4 can be used as a starting model for other local optimization methods, such as the L-BFGS method. RESULTS We validate the proposed hybrid optimization method for FWI on a synthetic 2D Marmousi model (Figure 1). The model consists of a complex subsurface structure with varying velocities and density. We simulate the elastic wave propagation using the finite-difference method. The source wavelet is a Ricker wavelet with a peak frequency of 10 Hz. We place 121 receivers along the surface with a spacing of 50 m, and record the wavefield at every time step. The recorded data are used as the observed data in the inversion. We compare the performance of the proposed hybrid optimization method with the local optimization method based on the L-BFGS algorithm. We use the L2 norm misfit function as the objective function in both methods. The starting model for the inversion is a simple layered model with constant velocities. We perform the inversion using both methods with the same computational resources and stopping criteria. Figure 2 shows the inverted velocity models using the two methods. The model obtained with the hybrid optimization method shows a better match with the true model than the one obtained with the L-BFGS method. The hybrid optimization method is also able to recover more details in the subsurface structure than the L-BFGS method. The misfit between the observed data and the estimated data is lower for the hybrid optimization method than for the L-BFGS method (Figure 3). The computational cost of the hybrid optimization method is slightly higher than that of the L-BFGS method, due to the additional global optimization step with VFSA. However, the hybrid optimization method is able to generate an accurate background velocity model, which reduces the dependence of the inversion on the choice of the starting model, and improves the robustness of the inversion. CONCLUSION We propose a hybrid optimization method for FWI that combines the VFSA global optimization algorithm with a derivativebased local optimization method. The method is able to generate an accurate background velocity model and is less sensitive to the choice of the starting model than traditional local optimization methods. The output of the method can be used as a starting model for further refinement using other local optimization methods. We demonstrate the effectiveness of the method on a synthetic 2D Marmousi model and show that it is able to recover more details in the subsurface structure than a traditional local optimization method. Future work includes testing the method on real field datasets and exploring the use of other global optimization methods