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new method for wave propagation modeling is introduced in this paper. By using the constraint optimization (Lagrange multiplier) method, the sum of weighted squared Fourier amplitudes is minimized when subjected to a constraint. The sum of the maximum amplitudes obtained from all output models is normalized to unity and is taken as a constraint. In this method, all the actual time histories are considered as outputs and dealt with equally. Independently of the combinations of time histories (or the first time history selected) during the analysis, the method captures the relationship of actual time histories by showing clear peaks. This paper describes the formulation of the models and illustrates the advantage of this method over the normalized input-output minimization (NIOM) method. The Mod-NIOM is then used to analyze the time histories of the Hyogoken-nanbu earthquake recorded at the Port Island vertical array site in Kobe, which suffered from liquefaction caused by the strong motions during the main shock. This method showed good correlations between the observed time histories at the site even though the surface time history was greatly modified by the liquefaction. [ABSTRACT FROM AUTHOR]