Bayesian Gls Regression For Regionalization Of Hydrologic Statistics Floods And Bulletin 17 Skew

The research presented in this thesis develops new statistical techniques for estimating regional skewness coefficients to improve flood frequency analysis in the United States. Flood frequency guidelines for the United States, specified in Bulletin 17B, recommend fitting the log-Pearson Type III (LP3) distribution to the series of annual flood maxima, in which the third moment of the distribution, the skewness coefficient , is combined with a regional skewness coefficient to improve its precision. The research presented here extends the quasi-analytic Bayesian analysis of the Generalized Least Squares (GLS) regional hydrologic regression framework introduced by Reis et al. [2005] to more accurately and precisely estimate regional skewness coefficients. Specifically, formulas derived within a Bayesian regression framework for the computation of estimators, standard errors, and diagnostic statistics are provided by Reis [2005] and Reis et al. [2005]. Diagnostic statistics further developed here include a Bayesian plausibility value, pseudo adjusted R-squared, pseudo-Analysis of Variance table, two diagnostic error variance ratios, as well as leverage and influence metrics. In addition, this research also develops a new influence diagnostic statistic which, in conjunction with the Bayesian extension of GLS leverage and influence metrics, can be used to better identify rogue observations and to effectively address lack-of-fit when estimating skewness coefficients. Currently, Bulletin 17B allows for regional skew values to be obtained from the skew map included with the Bulletin. As it is over 30 years old, the regional skew values from the Bulletin 17B skew map do not reflect annual maximum data acquired since 1976. This increase in available data, along with advances in computing power to support the Bayesian GLS regional hydrologic regression framework, allow for a much more precise estimate of the regional skewness coefficient for use in flood frequency analysis. This research employs the Bayesian GLS regression framework to estimate regional log-space skewness coefficients for three data sets: the Illinois River basin, the state of South Carolina, and the Southeastern United States. Bulletin 17B allows for the generation of skew prediction equations as an alternative method for determining regional skew coefficients when the mean squared error of the equations is smaller than reported from the Bulletin's skew map. These skew prediction equations can be generated using Ordinary Least Squares analysis, Weighted Least Squares analysis, Generalized Least Squares analysis employing the method of moment model-error-variance estimator introduced by Stedinger and Tasker [1985, 1986ab], or the new Bayesian GLS estimator. The advantages of using the Bayesian GLS estimation technique to determine a skew prediction equation are demonstrated here in the Illinois River basin and the state of South Carolina studies. To correctly analyze the Southeastern United States data set, methods are developed for identifying and screening redundant sites corresponding to nested watersheds with similar drainage areas. Special attention is devoted to developing an improved cross-correlation model of annual peak flows. The Bayesian GLS analysis using 342 stations from the Southeastern U.S. results in a highly accurate, constant regional skew model, with an average variance of prediction equal to 0.14. More complex models which include regional information and basin characteristics as additional regression parameters result in very little improvement. The application of the Bayesian estimator in the Southeastern study generates improved results over the mean square error of 0.30 reported for the Bulletin 17B regional map skew.

The research presented in this dissertation develops new statistical techniques for estimating regional relationships of hydrologic statistics. These techniques include extensions of the quasi-analytic Bayesian Generalizes Least Squares (B-GLS) framework presented in Reis et al. [2005] and further developed by Gruber et al. [2007] and Gruber and Stedinger [2008]. Recent extensions include a Pseudo [R squared sub delta] and pseudo analysis of variance table, plus a range of model performance, diagnostic and goodness-of-fit statistic. This dissertation develops a more stable Bayesian WLS/GLS procedure with the corresponding measures of precision and model performance. Special attention is given to model performance criteria, and the meaning of and insight provided by alternative measures of leverage and influence. Examples address development of regional skewness coefficients to improve flood frequency analysis in the United States. Large cross-correlations between annual peak discharges, coupled with relatively small model error variances, present difficulties in regional GLS skewness analyses. The B-GLS framework seeks to exploit the cross-correlations among the sample skewness estimates to obtain the best possible estimates of the model parameters. However, if the cross-correlations are large, the GLS estimators can become relatively complicated as a result of the effort to find the most efficient estimator of the parameters. Unfortunately, it appears that the precision of the cross-correlation estimates between any two particular sites is not of sufficient precision to justify the seemingly incorrect weights (both positive and negative) that the B-GLS analysis generates. Thus, an alternate regression procedure using both Weighted Least Squares (WLS) and GLS is developed so that the regional skewness analysis can provide both stable and defensible results. This alternate regression framework, is applied to two different data sets from different parts of the United States: the State of California and the Southeastern United States, to develop regional skewness estimators for flood frequency analysis. In addition, special attention is devoted to comparing and developing leverage and influence diagnostics statistics for GLS and WLS/GLS analyses, which can be used to identify rogue observations and to effectively address lack-of-fit when estimating hydrologic statistics.

The thesis here reports on and expands the results published in Lamontagne et al. [2012]. A hybrid Bayesian weighted/generalized least squares regression procedure is used to generate regional skew models for annual maximum rainfall floods of various durations in California. The procedure uses weighted least squares to estimate the model coefficients, and generalized least squares to estimate model precision. This procedure is necessitated by the unusually high cross-correlation exhibited between concurrent rainfall floods at different sites, which caused the regression weights to become unjustifiably erratic. New diagnostic statistics are developed for this special case and applied to real data. Overall model precision is excellent, which is important in the context of Bulletin 17B flood frequency analysis. Chapter 1 of the thesis provides an introductory background to flood frequency analysis, and the scope and area of the study. Chapter 1 also describes the procedure used by the United States Army Corps of Engineers to develop the rainfall flood time series. Chapter 2 discusses the characteristics of the log-Pearson Type III distribution, the Bulletin 17B flood frequency procedure, the Expected Moments Algorithm, and the effect of outliers on frequency estimation and tests for their identification and removal. Chapter 3 describes the development of weighted least squares and generalized least squares for regionalization of hydrologic variables. Chapter 3 then derives the new hybrid weighted/generalized least squares regression procedure and its accompanying diagnostic statistics. Finally, Chapter 3 discusses recent research which uses an alternative generalized least squares framework. Chapter 4 details the application of the procedure from Chapter 3 to rainfall flood of various durations from California to create a regional skew model for California. Finally, Chapter 5 examines various aspects of the analysis in Chapter 4 which were noticeably different from previous regional skew studies. In particular, Chapter 4 reexamines the Pseudo ANOVA table and proposes a new, alternative table. ii.

The book presents the processes governing the dynamics of landscapes, soils and sediments, water and energy under different climatic regions using studies conducted in varied climatic zones including arid, semi-arid, humid and wet regions. The spatiotemporal availability of the processes and fluxes and their linkage to the environment, land, soil and water management are presented at various scales. Spatial scales including laboratory, field, watershed, river basin and regions are represented. The effect of tillage operations and land management on soil physical characteristics and soil moisture is discussed. The book has 35 chapters in seven sections: 1) Landscape and Land Cover Dynamics, 2) Rainfall-Runoff Processes, 3) Floods and Hydrological Processes 4) Groundwater Flow and Aquifer Management, 5) Sediment Dynamics and Soil Management, 6) Climate change impact on vegetation, sediment and water dynamics, and 7) Water and Watershed Management.

Flood-frequency information is important in the Central Valley region of California because of the high risk of catastrophic flooding. Most traditional flood-frequency studies focus on peak flows, but for the assessment of the adequacy of reservoirs, levees, other flood control structures, sustained flood flow (flood duration) frequency data are needed. This study focuses on rainfall or rain-on-snow floods, rather than the annual maximum, because rain events produce the largest floods in the region. A key to estimating flood-duration frequency is determining the regional skew for such data. Of the 50 sites used in this study to determine regional skew, 28 sites were considered to have little to no significant regulated flows, and for the 22 sites considered significantly regulated, unregulated daily flow data were synthesized by using reservoir storage changes and diversion records. The unregulated, annual maximum rainfall flood flows for selected durations (1-day, 3-day, 7-day, 15-day, and 30-day) for all 50 sites were furnished by the U.S. Army Corps of Engineers. Station skew was determined by using the expected moments algorithm program for fitting the Pearson Type 3 flood-frequency distribution to the logarithms of annual flood-duration data. Bayesian generalized least squares regression procedures used in earlier studies were modified to address problems caused by large cross correlations among concurrent rainfall floods in California and to address the extensive censoring of low outliers at some sites, by using the new expected moments algorithm for fitting the LP3 distribution to rainfall flood-duration data. To properly account for these problems and to develop suitable regional-skew regression models and regression diagnostics, a combination of ordinary least squares, weighted least squares, and Bayesian generalized least squares regressions were adopted. This new methodology determined that a nonlinear model relating regional skew to mean basin elevation was the best model for each flood duration. The regional-skew values ranged from -0.74 for a flood duration of 1-day and a mean basin elevation less than 2,500 feet to values near 0 for a flood duration of 7-days and a mean basin elevation greater than 4,500 feet. This relation between skew and elevation reflects the interaction of snow and rain, which increases with increased elevation. The regional skews are more accurate, and the mean squared errors are less than in the Interagency Advisory Committee on Water Data's National skew map of Bulletin 17B.