Due to the influence of variations in landform, geophysical data acquisition is usually subsampled. Reconstruction of the seismic wavefield from subsampled data is an ill-posed inverse problem. Compressive sensing (CS) can be used to recover the original geophysical data from the subsampled data. In this paper, we consider the wavefield reconstruction problem as a CS and propose a piecewise random subsampling scheme based on the wavelet transform. The proposed sampling scheme overcomes the disadvantages of uncontrolled random sampling. In computation, an l1-norm constrained trust region method is developed to solve the CS problem. Numerical results demonstrate that the proposed sampling technique and the trust region approach are robust in solving the ill-posed CS problem and can greatly improve the quality of wavefield recovery.