In fluorescence molecular tomography (FMT), the fluorophore distribution is reconstructed using the diffuse-light measurements obtained from the rotating source-detector pairs placed on the boundary of the tissues. Owing to the intensity attenuation of light when it propagates through tissues, the sensitivity of measurements deteriorates quickly with increased depth. Thus the inconsistent contrast of reconstructed fluorophores located at different depths is a major challenge in FMT. As a spatially variant regularization method, the adaptive support driven reweighted L1-minimization (ASDR-L1) algorithm is proposed here for depth compensation in FMT. ASDR-L1 is a modification of the restarted L1 regularization-based nonlinear conjugate gradient (re-L1-NCG) algorithm previously proposed by our laboratory. In ASDR-L1, the original L1-minimization problem is replaced by a sequence of weighted L1-minimization subproblems with spatially updated weights applied to the adaptive support estimate. Like re-L1-NCG, ASRDR-L1 adopts the restarted strategy in each outer iteration, which contributes to the adaptive support estimate. The updated weights for the next iteration spatially depend on the current solution. In the support estimate, spatially updated weights mean different regularization parameters for different locations. A large regularization parameter in the weighted L1-minimization subproblem makes the results concentrate on a small number of large values, whereas a small regularization parameter tends to make the values be evenly distributed. Thus depth compensation in FMT is achieved through the iteratively updated weights. Simulation experiments are conducted to confirm the feasibility of ASDR-L1. Through ASDR-L1, the reconstructed contrast between two identical fluorophores located at different depths is increased from 1:0.43 to 1:0.96.