As a new development form for evaluating the regional water resources carrying capacity, forewarning regional water resources of their carrying capacities is an important adjustment and control measure for regional water security management. Up to now, most research on this issue have been qualitative analyses, with a lack of quantitative research. For this reason, an index system for forewarning regional water resources of their carrying capacities and grade standards, has been established in Anhui Province, China, in this paper. Subjective weights of forewarning indices can be calculated using a fuzzy analytic hierarchy process, based on an accelerating genetic algorithm, while objective weights of forewarning indices can be calculated by using a projection pursuit method, based on an accelerating genetic algorithm. These two kinds of weights can be combined into combination weights of forewarning indices, by using the minimum relative information entropy principle. Furthermore, a forewarning model of regional water resources carrying capacity, based on entropy combination weight, is put forward. The model can fully integrate subjective and objective information in the process of forewarning. The results show that the calculation results of the model are reasonable and the method has high adaptability. Therefore, this model is worth studying and popularizing.

This paper establishes a water resources vulnerability framework based on sensitivity, natural resilience and artificial adaptation, through the analyses of the four states of the water system and its accompanying transformation processes. Furthermore, it proposes an analysis method for water resources vulnerability based on connection entropy, which extends the concept of contact entropy. An example is given of the water resources vulnerability in Anhui Province of China, which analysis illustrates that, overall, vulnerability levels fluctuated and showed apparent improvement trends from 2001 to 2015. Some suggestions are also provided for the improvement of the level of water resources vulnerability in Anhui Province, considering the viewpoint of the vulnerability index.

The adjoint function of connection number has unique advantages in solving uncertainty problems of water resource complex systems, and has become an important frontier and research hotspot in the uncertainty research of water resource complex problems. However, in the rapid evolution of the adjoint function, some problems greatly limit the application of the adjoint function in the research of water resources. Therefore, based on bibliometric analysis, development, practical application issues, and prospects of the hot directions are analyzed. It is found that the development of the connection number of water resource set pair analysis can be divided into three stages: (1) relatively sluggish development before 2005, (2) a period of rapid advancement in adjoint function research spanning from 2005 to 2017, and (3) a subsequent surge post-2018. The introduction of the adjoint function of connection number promotes the continuous development of set pair analysis of water resources. Set pair potential and partial connection number are the crucial research directions of the adjoint function. Subtractive set pair potential has rapidly developed into a relatively independent and important trajectory. The research on connection entropy is comparatively less, which needs to be further strengthened, while that on adjacent connection number is even less. The adjoint function of set pair potential can be divided into three major categories: division set pair potential, exponential set pair potential, and subtraction set pair potential. The subtraction set pair potential, which retains the original dimension and quantity variation range of the connection number, is widely used in water resources and other fields. Coupled with the partial connection number, a series of new connection number adjoint functions have been developed. The partial connection number can be mainly divided into two categories: total partial connection number, and semi-partial connection number. Among these, the calculation expression and connotation of total partial connection numbers have not yet reached a consensus, accompanied by the slow development of high-order partial connection numbers. Semi-partial connection number can describe the mutual migration movement between different components of the connection number, which develops rapidly. With the limitations and current situation described above, promoting the exploration and application of the adjoint function of connection number in the field of water resources and other fields of complex systems has become the focus of future research.

Variability identification and uncertainty characteristic analysis, under the impacts of climate change and human activities, is beneficial for accurately predicting the future evolution trend of hydrological variables. In this study, based on the evolution trend and characteristic analyses of historical precipitation and temperature sequences from monthly, annual, and interannual scales through the Linear Tendency Rate (LTR) index, as well as its variability point identification using the M–K trend test method, we further utilized three cloud characteristic parameters comprising the average Ex, entropy En, and hyper-entropy He of the Cloud Model (CM) method to quantitatively reveal the uncertainty features corresponding to the diverse cloud distribution of precipitation and temperature sample scatters. And then, through an application analysis of the proposed research framework in Anhui Province, China, the following can be summarized from the application results: (1) The annual precipitation of Anhui Province presented a remarkable decreasing trend from south to north and an annual increasing trend from 1960 to 2020, especially in the southern area, with the LTR index equaling 55.87 mm/10a, and the annual average temperature of the entire provincial area also presented an obvious increasing trend from 1960 to 2020, with LTR equaling about 0.226 °C/10a. (2) The uncertainty characteristic of the precipitation series was evidently intensified after the variability points in 2013 and 2014 in the southern and provincial areas, respectively, according to the derived values of entropy En and hyper-entropy He, which are basically to the contrary for the historical annual average temperature series in southern Anhui Province. (3) The obtained result was basically consistent with the practical statistics of historical hydrological and disaster data, indicating that the proposed research methodologies can be further applied in related variability diagnosis analyses of non-stationary hydrological variables.