We address the existence of surface solitons at an interface in a defocnsing cubic medium with an imprinted one-dimensional (1D) composite Bessel optical lattice. This setting is composed of two Bessel lattices with different orders and different modulation depths, separated beside both sides of an interface. Stability analysis and numerical propagation simulations prove that solitons supported by the model are dynamically stable in the entire domain of their existence. The order of lattice determines the shape of soliton, and the amplitude of soliton depends on the lattice modulation depth.The experimental realization of the scheme is also proposed.Our results may provide another effective way of controlling the shapes of surface solitons and thus their evolutions by introducing a new freedom degree.