Differential semblance measures the deviation from flatness or focus of image gathers. The differential semblance objective function posed on the sub-surface offset domain responds smoothly to velocity changes. Therefore gradient descent methods are uniquely attractive for velocity updating by differential semblance optimization. Because of their kinematic fidelity, wave equation (depth extrapolation) migration methods are natural platforms for velocity analysis in complex structures. The gradient of the objective function with respect to velocity is fomulated through the adjoint of differential migration. Limited memory BFGS algorithm is used for the velocity optimization. The method for wave equation velocity analysis developed in this thesis study is applied to both synthetic and real data examples.