The horizontal and vertical extended impulse response Gramians (EIRG) of causal recursive separable denominator (CRSD) 2-D digital systems are defined in this paper. Based on the 2-D EIRG, a new identification and model reduction algorithm is presented for 2-D CRSD systems. It is shown that the reduced-order system has three important properties. First, the 2-D EIRG of the reduced-order 2-D system approximate the 2-D EIRG of the original 2-D system. Second, the 2-D system obtained is a suboptimal solution to the minimization of the energy of the 2-D error system between the original 2-D system and the reduced-order system. Finally, the structure of the reduced-order 2-D system is convergent to a balanced realization. Furthermore, the present algorithm can handle impulse response data of deterministic or stochastic nature for 2-D system identification application.