When such plans and specifications have been prepared and the estimates of the cost and expense of making the improvement have been adopted by the council and when an estimate has been made of the compensation to be paid for property damaged or taken, either before or after the compensation has been ascertained in the eminent domain proceedings, the city council shall cause an assessment roll to be prepared containing a list of all of the property within the improvement district which it is proposed to assess for the improvement, together with the names of the owners, if known, and if unknown the property shall be assessed to an unknown owner, and opposite each description shall be set the amount assessed to such description.

When so ordered by the council, the entire amount of compensation paid or to be paid for property damaged or taken, including all of the costs and expenses incidental to the condemnation proceedings together with the entire cost and expense of making the improvement, may be assessed against the property within the district subject to assessment, but the council may order any portion of the costs paid out of the current or general expense fund of the city.

The assessments shall be made according to and in proportion to surface area one square foot of surface to be the unit of assessment, except that the several parcels of land in any enlarged district not actually filled shall be assessed in accordance with special benefits: PROVIDED, That where any parcel of land was partially filled by the owner prior to the initiation of the improvement, an equitable deduction for such partial filling may be allowed.

The cost and expense incidental to the filling of the streets, alleys and public places within such assessment district shall be borne by the private property within such district subject to assessment when so ordered by the council. When the assessments are payable in installments, the assessment roll when equalized, shall show the number of installments and the amounts thereof. The assessments may be made payable in any number of equal annual installments not exceeding ten in number.