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Using linearization, we reformulate the system of PDEs for the time-to-depth conversion and interval velocity estimation process to a simpler system appropriate for handling weak lateral variations. The new system can be solved in a downward continuation fashion with a significantly improved computational efficiency. . Our numerical examples show that the proposed method produces accurate results that honor the effects of lateral heterogeneity with a speedup by an order of magnitude. Therefore, the results of the proposed method can be used to correct the conventional Dix conversion and to produce a starting velocity model for more accurate velocity-estimation methods.