The timing of DNA replication is largely regulated by the location and timing of replication origin firing. Therefore, much effort has been invested in identifying and analyzing human replication origins. However, the heterogeneous nature of eukaryotic replication kinetics and the low efficiency of individual origins in metazoans has made mapping the location and timing of replication initiation in human cells difficult. We have mapped early-firing origins in HeLa cells using Optical Replication Mapping, a high-throughput single-molecule approach based on Bionano Genomics genomic mapping technology. The single-molecule nature and 290-fold coverage of our dataset allowed us to identify origins that fire with as little as 1% efficiency. We find sites of human replication initiation in early S phase are not confined to well-defined efficient replication origins, but are instead distributed across broad initiation zones consisting of many inefficient origins. These early-firing initiation zones co-localize with initiation zones inferred from Okazaki-fragment-mapping analysis and are enriched in ORC1 binding sites. Although most early-firing origins fire in early-replication regions of the genome, a significant number fire in late-replicating regions, suggesting that the major difference between origins in early and late replicating regions is their probability of firing in early S-phase, as opposed to qualitative differences in their firing-time distributions. This observation is consistent with stochastic models of origin timing regulation, which explain the regulation of replication timing in yeast.