In this letter, a new sparsity-promoting penalty function is introduced for sensor selection problems in field reconstruction, which has the property of avoiding scenarios where the same sensors are successively selected. Using a reweighted ℓ₁ relaxation of the ℓ₀ norm, the sensor selection problem is reformulated as a convex quadratic program. In order to handle large-scale problems, we also present two fast algorithms: accelerated proximal gradient method and alternating direction method of multipliers. Numerical results are provided to demonstrate the effectiveness of our approaches.