A robust data assimilation scheme is presented for a wave model to predict evolving nonlinear ocean waves. The Fourier coefficients of the surface elevation and the free surface velocity potential are chosen for state variables and are propagated in time by solving numerically a set of nonlinear evolution equations using a pseudo-spectral method. The numerical solutions are then updated with noise corrupted measurements of the surface elevation with the aid of an explicit Kalman filter for which the time evolution of the error covariance matrix is found explicitly by solving analytically the linearized wave prediction model. After presenting an error analysis for this explicit data assimilation scheme, numerical simulations of the integrated nonlinear wave prediction model for long-crested waves of varying wave steepness are performed by using synthetic data with different noise characteristics. It is shown that the estimated surface wave fields agree well with the true states, and the present data assimilation scheme based on the explicit Kalman filter improves considerably the computational efficiency and stability, in comparison with a standard Kalman filter for which the error covariance matrix is found numerically.