In the event payment is to be made by an additional assessment, the additional assessment shall be made and collected, and notice thereof shall be given in the same manner, as nearly as may be, as in the case of the original assessment. The superintendent of streets shall prepare and file with the additional assessment, a diagram showing the lots or parcels of land deemed by him to have been benefited by the work. The superintendent of streets shall assess the additional assessment upon the several lots or parcels of land shown in such diagram in the proportion to the estimated benefits to be received by each of the several lots or parcels of land.