Drought propagation and construction of a comprehensive drought index based on the Soil and Water Assessment Tool (SWAT) and empirical Kendall distribution function (<i>K</i>_C′): a case study for the Jinta River basin in northwestern China

Monitoring drought and mastering the laws of drought propagation are the basis for regional drought prevention and resistance. Multivariate drought indicators considering meteorological, agricultural and hydrological information may fully describe drought conditions. However, series of hydrological variables in cold and arid regions that are too short or missing make it difficult to monitor drought. This paper proposed a method combining Soil and Water Assessment Tool (SWAT) and empirical Kendall distribution function (KC′) for drought monitoring. The SWAT model, based on the principle of runoff formation, was used to simulate the hydrological variables of the drought evolution process. Three univariate drought indexes, namely meteorological drought (standardized precipitation evapotranspiration index; SPEI), agricultural drought (standardized soil moisture index; SSI) and hydrological drought (standardized streamflow drought index; SDI), were constructed using a parametric or non-parametric method to analyze the propagation time from meteorological drought to agricultural drought and hydrological drought. The KC′ was used to build a multivariable comprehensive meteorology–agriculture–hydrology drought index (MAHDI) that integrated meteorological, agricultural and hydrological drought to analyze the characteristics of a comprehensive drought evolution. The Jinta River in the inland basin of northwestern China was used as the study area. The results showed that agricultural and hydrological drought had a seasonal lag time from meteorological drought. The degree of drought in this basin was high in the northern and low in the southern regions. MAHDI proved to be acceptable in that it was consistent with historical drought records, could catch drought conditions characterized by univariate drought indexes, and capture the occurrence and end of droughts. Nevertheless, its ability to characterize mild and moderate droughts was stronger than severe droughts. In addition, the comprehensive drought conditions showed insignificant aggravating trends in spring and summer and showed insignificant alleviating trends in autumn and winter and at annual scales. The results provided theoretical support for the drought monitoring in the Jinta River basin. This method provided the possibility for drought monitoring in other watersheds lacking measured data.

BIVARIATE MARSHALL–OLKIN EXPONENTIAL SHOCK MODEL

The well-known Marshall–Olkin model is known for its extension of exponential distribution preserving lack of memory property. Based on shock models, a new generalization of the bivariate Marshall–Olkin exponential distribution is given. The proposed model allows wider range tail dependence which is appealing in modeling risky events. Moreover, a stochastic comparison according to this shock model and also some properties, such as association measures, tail dependence and Kendall distribution, are presented. The new shock model is analytically quite tractable, and it can be used quite effectively, to analyze discrete–continuous data. This has been shown on real data. Finally, we propose the multivariate extension of the Marshall–Olkin model that has some intersection with the well-known multivariate Archimax copulas.

Relations between ageing and dependence for exchangeable lifetimes with an extension for the IFRA/DFRA property

We first review an approach that had been developed in the past years to introduce concepts of “bivariate ageing” for exchangeable lifetimes and to analyze mutual relations among stochastic dependence, univariate ageing, and bivariate ageing.A specific feature of such an approach dwells on the concept of semi-copula and in the extension, from copulas to semi-copulas, of properties of stochastic dependence. In this perspective, we aim to discuss some intricate aspects of conceptual character and to provide the readers with pertinent remarks from a Bayesian Statistics standpoint. In particular we will discuss the role of extensions of dependence properties. “Archimedean” models have an important role in the present framework.In the second part of the paper, the definitions of Kendall distribution and of Kendall equivalence classes will be extended to semi-copulas and related properties will be analyzed. On such a basis, we will consider the notion of “Pseudo-Archimedean” models and extend to them the analysis of the relations between the ageing notions of IFRA/DFRA-type and the dependence concepts of PKD/NKD.

Multivariate return periods in hydrology: a critical and practical review focusing on synthetic design hydrograph estimation

Most of the hydrological and hydraulic studies refer to the notion of a return period to quantify design variables. When dealing with multiple design variables, the well-known univariate statistical analysis is no longer satisfactory, and several issues challenge the practitioner. How should one incorporate the dependence between variables? How should a multivariate return period be defined and applied in order to yield a proper design event? In this study an overview of the state of the art for estimating multivariate design events is given and the different approaches are compared. The construction of multivariate distribution functions is done through the use of copulas, given their practicality in multivariate frequency analyses and their ability to model numerous types of dependence structures in a flexible way. A synthetic case study is used to generate a large data set of simulated discharges that is used for illustrating the effect of different modelling choices on the design events. Based on different uni- and multivariate approaches, the design hydrograph characteristics of a 3-D phenomenon composed of annual maximum peak discharge, its volume, and duration are derived. These approaches are based on regression analysis, bivariate conditional distributions, bivariate joint distributions and Kendall distribution functions, highlighting theoretical and practical issues of multivariate frequency analysis. Also an ensemble-based approach is presented. For a given design return period, the approach chosen clearly affects the calculated design event, and much attention should be given to the choice of the approach used as this depends on the real-world problem at hand.