Application of uncertainty and reliability analysis is an essential part of many problems related to modeling and decision making in the area of environmental and water resources engineering. Computational efficiency, understandability, and easier application have made the first-order approximation (FOA) method a favored tool for uncertainty analysis. In many instances, situations may arise where the accuracy of FOA estimates becomes questionable. Often the FOA application is considered acceptable if the coefficient of variation of the uncertain parameter(s) is <0.2, but this criterion is not correct in all the situations. Analytical as well as graphical relations for relative error are developed and presented for a generic power function that can be used as a guide for judging the suitability of the FOA for a specified acceptable error of estimation. Further, these analytical and graphical relations enable FOA estimates for means and variances of model components to be corrected to their true values. Using these corrected values of means and variances for model components, one can determine the exact values of the mean and variance of an output random variable. This technique is applicable when an output variable is a function of several independent random variables in multiplicative, additive, or in combined (multiplicative and additive) forms. Two examples are given to demonstrate the application of the technique. ¿ 2001 American Geophysical Union