The equivalence between two quaternion based coarse alignment algorithms for strapdown inertial navigation system (SINS) with inertial frame is carefully analyzed. Firstly, the alignment of SINS is transformed into the Wahba problem in attitude determination based on the decomposition principle of the attitude matrix, and an integral form objective function to be optimized is established according to the fact that the gravity vector changes continuously. Then the objective function is transformed into the function of attitude quaternion on the basis of coordinate transformation rules in the quaternion form, and two algorithms are derived to acquire the optimal attitude quaternion. Finally the equivalence between the two algorithms is demonstrated based on the properties of matrix eigenvalues and eigenvectors, with further verification conducted with data from both simulation and field experiment.