## 1. Introduction

[2] Regional frequency analysis (RFA) gained recognition as a viable option to arrive at design estimates of variables associated with hydrometeorological events such as extreme precipitation and floods at target locations in river basins that are ungauged or have limited records. The analysis involves (i) use of a regionalization approach for identification of locations that are similar to the target location (site), in terms of mechanisms influencing the variable being analyzed, to form a homogeneous region, and (ii) use of a RFA approach to fit a distribution to information pooled from the region for arriving at design estimate. Among the various RFA approaches that have been developed in the past, index-flood approach [*Dalrymple*, 1960] gained wide recognition, which makes the following assumptions: (i) records of the variable at each site in a region are identically distributed; (ii) records at each site are serially independent; (iii) there is no dependence between records at different sites; and (iv) frequency distribution of the variable is identical across sites in the region, except for a site-specific scaling factor called index-flood. Of these assumptions, the first three are generally valid for analysis of a random variable representing hydrometeorological extreme event, but the fourth is specific to only index-flood approach. Implementation of the index-flood approach involves normalization of records of the variable for each site by dividing them by the site's scaling factor and combining information from those normalized records to construct a “dimensionless distribution function” (growth curve) that is assumed to be unique for all the sites in the region. Required quantiles at the target site are estimated by multiplying the growth curve by site-specific scaling factor, which is often chosen as mean of the variable. Alternate scaling factors that have been considered in previous studies include median, trimmed mean, and quantile of the at-site distribution [*Smith*, 1989; *Hosking and Wallis*, 1997].

[3] For the index-flood approach to be effective, the aforementioned assumptions (i)–(iv) should be valid for the records before and after normalization. Validity of the first three assumptions can be ensured by considering the scaling factor to be a population statistic. However, as population statistic is unknown in real-world scenario, modelers chose sample statistic for normalization. As a result, frequency distribution of record at each site undergoes a change and the assumptions (ii) and (iii) would be violated. These effects of normalization were reported in a number of previous studies [e.g., *Stedinger*, 1983; *Hosking and Wallis*, 1997, p. 88; *Sveinsson et al*., 2001, 2003]. *Stedinger* [1983] attempted to overcome the problem by implementing the index-flood approach in log-space and suggested use of unbiased moment or probability weighted moment estimators. *Boes et al*. [1989] suggested use of the method of maximum likelihood to estimate parameters of growth curve assuming regional frequency distribution to be Weibull. *Sveinsson et al*. [2001] extended this approach to generalized extreme value (GEV) distribution and termed it as population index-flood (PIF) method. Literature review on different aspects related to the index-flood approach can be found in *Stedinger and Lu* [1995] and *Bocchiola et al*. [2003].

[4] For situations where sample statistic (mean) is chosen as a scaling factor, *Sveinsson et al*. [2001] proved through analytical formulations that assumption (ii) of index-flood approach (i.e., independence of records at each site) would be invalid, and frequency distribution of the normalized data would be different from that of the original data. Further, it was argued that the fourth assumption of index-flood approach (i.e., distributions of normalized records would be identical for sites) would be invalid if record lengths of sites in a homogeneous region are different. In real-world scenario, the scale and shape parameters of sites in a homogeneous region may not be close enough to be considered identical, even if the type of frequency distribution is the same for all the sites in the region. It can even be theoretically established that probability of those parameter values being exactly equal for any two sites in a region is zero, even if those sites have unlimited record lengths. The shortcomings associated with index-flood approach motivated development of an alternate mathematical approach to RFA in this paper. The RFA is deemed to be effective if knowledge of location, scale, as well as shape parameters of all the sites is utilized in the analysis, to properly characterize the growth curve (dimensionless distribution function) that represents the region. The proposed approach (PA) involves (i) identification of an appropriate frequency distribution to fit the random variable being analyzed for the homogeneous region, (ii) use of a proposed transformation mechanism to map observations of the variable from original space to a dimensionless space where the form of distribution does not change, and variation in values of location, scale, as well as shape parameters of the distribution is minimal across sites, thus satisfying all the assumptions of index-flood approach, (iii) construction of a growth curve in the dimensionless space, and (iv) mapping the growth curve to the original space for the target site by applying proposed inverse transformation to arrive at required quantile(s) for the site.

[5] The reminder of this paper is structured as follows: index-flood approach and the problem being addressed in this paper are described in section 2. Following that methodology for proposed RFA approach is presented in section 3. Subsequently effectiveness of the proposed methodology is demonstrated through Monte Carlo simulation experiments and by application to real-world data in section 4. Finally, summary and concluding remarks are given in section 5.