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Routing in multihop wireless networks is challenging, mainly due to unreliable wireless links/channels. Geographic opportunistic routing (GOR) was proposed to cope with the unreliable transmissions by exploiting the broadcast nature of the wireless medium and the spatial diversity of the network topology. Previous studies on GOR have focused on networks with a single channel rate. The capability of supporting multiple channel rates, which is common in current wireless systems, has not carefully been studied for GOR. In this paper, we carry out a study on the impacts of multiple rates, as well as candidate selection, prioritization, and coordination, on the performance of GOR. We propose a new local metric, i.e., the opportunistic effective one-hop throughput (OEOT), to characterize the tradeoff between one-hop packet advancement and packet forwarding time. We further propose a local rate adaptation and candidate-selection algorithm to approach the optimum of this metric. The simulation results show that the multirate GOR (MGOR) incorporating the rate adaptation and candidate-selection algorithm achieves higher throughput and lower delay than the corresponding single-rate and multirate traditional geographic routing and single-rate opportunistic routing protocols.