This article investigates the problem of secure state estimation for continuous-time linear systems in the presence of sparse sensor attacks. Compared with the existing results, the attacked sensor set can be changed by adversaries against secure estimation. To address the more erratic attacks, a novel supervisory state observer is proposed, which employs a bank of candidate nonlinear subobservers and a switching logic administrated by a monitoring function to select the active subobserver at every instant of time. Based on the stability analysis of switched systems, it is proven that the supervisory observer asymptotically converges to a neighborhood of the true system state in the presence of the sensor attacks and bounded disturbances. A simulation example is given to substantiate the theoretical results.