Introduction

Earth and environmental sciences are vital for human survival and societal development. Natural resource management and environmental protection are becoming increasingly important (Li et al. 2019a, b, c). All resources needed for human survival and societal development, such as mineral resources, water resources (Li et al. 2018a, b; He et al. 2019a, b), land resources (He and Wu 2019a; Li and Qian 2018), and atmospheric resources, are derived from the earth, and seeking effective and sustainable ways of using these resources is crucial. Unreasonable exploitation and use of natural resources have resulted in a series of ecological and environmental problems, threatening human health and safety and requiring humans to take measures to reduce the risks induced by these problems (Li and Wu 2019). In this context, various decision-making optimization methods, as well as risk assessment and prediction approaches, have been proposed and applied in earth and environmental sciences. For example, the hierarchical cluster analysis (HCA) method (Wu et al. 2019), the fuzzy method (Chen et al. 2018; He and Wu 2019b), the projection pursuit method, the technique for order of preference by similarity to ideal solution (TOPSIS) method (Li et al. 2018c), matter-element extension analysis (Li et al. 2018d), and artificial neural networks have been applied in the assessment and prediction of groundwater quality. These approaches have assisted decision makers in finding the best solutions to environmental and engineering problems in the past several decades. However, human cognition of nature is still in a fuzzy state in some fields because of an incomplete understanding of the earth. Hence, set pair analysis (SPA) has been proposed to solve the problem of uncertainty by establishing a dynamic model.

Set pair analysis is a systematic mathematical theory which was first described 30 years ago by Zhao (1989) at a national symposium on system theory and regional planning. This approach attempts to explore the deterministic and uncertain relations of two sets and their interactions from the dimensions of “identity,”“discrepancy,” and “contrary.” The degree of the interaction is further described by the connection number, and the uncertainty can be addressed and ultimately analyzed by the calculation of the connection number (Su et al. 2019; Tian and Wu 2019). As such, SPA is also named connection mathematics. Benefiting from the interdisciplinary development, SPA has been used in a wide range of disciplines such as earth and environmental sciences, industrial production, military security, education, information security, transportation, healthcare, and public services (Yang and Gao 2016; Tsuei et al. 2020; Feng et al. 2018; Yang and Hu 2018; Bao and Zhang 2018), with the most rapid development in earth and environmental sciences. However, the development status and challenges of SPA in earth and environmental sciences are poorly understood, mainly because no comprehensive review on SPA application in these areas has been carried out, with the exception of some reviews in hydrology and water resources (Cui et al. 2018; Zhang et al. 2016). Previous studies have mainly summarized the development progress of SPA, but ignored the challenges, shortcomings, and future prospects of this tool. Therefore, this review focuses on these aspects in earth and environmental sciences. By summarizing and exploring the development status of SPA in earth and environmental sciences, this study enriches and improves the theoretical system of SPA and further promotes its development and application.

Basic Concept of SPA

Set pair analysis is an improved uncertainty theory which combines certainties with uncertainties as an integrated certainty–uncertainty system, systematically depicting certainty and uncertainty using the connection degree from the three aspects including identity, discrepancy, and contrary. For example, when there are two sets A and B, and both have N elements, among which S elements show identical properties, F elements demonstrate discrepant properties and Z elements display contrary properties. Then, the basic set pair H = (A, B) can be expressed using the following ternary connection number function:

$$\left\{ {\begin{array}{l} {\mu = \frac{S}{N} + \frac{F}{N}i + \frac{Z}{N}j= a + bi + cj}, \hfill \\ {a + b + c = 1}, \hfill \\ {a,\;b,\;c \in \left[ {0,\;1} \right]} ,\hfill \\ {i \in \left[ { - 1,\;1} \right]}, \hfill \\ {j = - 1}, \hfill \\ \end{array} } \right.$$
(1)

where μ represents the connection number, a, b, and c represent the degree of “identity,” “discrepancy,” and “contrary,” respectively, i is the uncertainty coefficient, and j is the contrary coefficient.

According to Eq. (1), the multi-element connection number can be expressed as follows:

$$\left\{ {\begin{array}{l} {\mu = a + b_{1} i_{1} + b_{2} i_{2} + \cdots + b_{t} i_{t} + cj,} \\ {a + b_{1} + b_{2} + \cdots + b_{t} + c = 1,} \\ {a,\;b_{1} ,\;b_{2} , \ldots ,b_{t} ,\quad c \in [0,\;1],} \\ {i_{p} \in \left[ { - 1 + \frac{{2(p - 1)}}{t},\; - 1 + \frac{{2p}}{t}} \right],\quad p = 1,\;2, \ldots ,t,} \\ {j = - 1,} \\ \end{array} } \right.$$
(2)

where b1i1 + b2i2 + ··· + btit represents the extension of “bi” in Eq. (1). As the core function of SPA is to deal with uncertainty problems, the connection number μ is not an exact value but a dynamic number, which is composed of the actual value a and the dynamic values bi and cj. Thus, the uncertainty relationships can be described in more detail and more comprehensively by SPA. In addition, the definitions of subtractive, multiplicative, and division operations of connection numbers have been put forward and the basic laws of operations have been studied (Huang et al. 2000). The unique advantage of SPA is the structural relationship of the connection number, which can solve a series of uncertainty problems and can also deal with the relation and transformation analysis between certainty and uncertainty by analyzing “identity,” “discrepancy,” and “contrary.”

In addition, the partial connection number is also a significant component of SPA and is divided into partial-positive connection number, partial-negative connection number, and partial connection number. As an adjoint function of the connection number, it can reflect the development trend of the research object. According to Eq. (1), the partial connection number can be obtained as follows:

Partial-positive connection number:

$$\partial \mu^{ + } = \frac{a}{a + b} + \frac{b}{b + c}i_{1} \quad \left( {i_{1} \in \left[ { - 1,\;1} \right]} \right) .$$
(3)

Partial-negative connection number:

$$\partial \mu^{ - } = \frac{b}{a + b} + \frac{c}{b + c}i_{2} \quad \left( {i_{2} \in \left[ { - 1,\;1} \right]} \right).$$
(4)

Partial connection number:

$$\partial \mu = \partial \mu^{ + } - \partial \mu^{ - } = \left( {\frac{a}{a + b} + \frac{b}{b + c}i_{1} } \right) - \left( {\frac{b}{a + b} + \frac{c}{b + c}i_{2} } \right) \quad \left( {i_{1} ,\;i_{2} \in \left[ { - 1,\;1} \right]} \right),$$
(5)

where ∂μ+ reflects the positive development trend of the research objects, ∂μ− reflects the negative development trend of the research objects, ∂μ represents the comprehensive development trend of the research objects, and i1 and i2 are the uncertainty coefficients. In general, the partial connection number can be used in establishing decision-making models. According to the value of ∂μ, the decisions can be divided into three categories: feasible (∂μ < 0.5), general (∂μ = 0.5), and infeasible (∂μ > 0.5), which means that the fit and unfit qualities of decision-making plans are inversely proportional to the value of ∂μ.

Compared to the traditional methods, SPA has unique advantages in that it cannot only be used as a systematic analysis method to treat uncertainty problems, but also reveals the relationship between integral and local structures, which can be shown in a dynamic formula. In addition, the theoretical system and the calculation method are simple and convenient for application. In actual research work, the research backgrounds including climate change, human activities, and the natural geographical environment are constantly changing, and there are intricate connections in the research system. Therefore, the corresponding set pairs can be formed in pairs, and SPA can be applied to solve the uncertainty relation of system analysis, prediction, evaluation, decision, optimization, simulation, reasoning, and regulation; the obtained results can be in line with the actual situation.

Development and Application of SPA in Earth and Environmental Sciences

SPA Application and Development in Engineering Geology

Set pair analysis has first been applied in engineering geology in 1997 by Gong (1997). The author preliminarily analyzed the problem of land subsidence in Shanghai and explored the restrictive relationship among social demand, resource status, and geological environment by using the SPA theory, which provided a new idea for solving geo-environmental problems. After Gong’s research, the SPA theory was gradually recognized by a few engineering geologists in the following 10 years. For example, Gao and Chen (2004) used the connection degree in project risk ranking, while Li et al. (2005) applied SPA in the coal mining safety capacity evaluation and prediction. These applications of SPA in engineering geology helped to expand the recognition of SPA by geologists and engineering geologists, benefiting the further development of SPA. Uncertainty problems of ground subsidence monitoring, foundation bearing capacity calculation and evaluation, as well as risk capacity evaluation could also be resolved by SPA (Wu et al. 2008a, b). Ultimately, scholars found that adopting SPA in engineering geology to solve typical engineering geological problems and reduce the uncertainties associated with these problems is feasible and reliable. In this area, SPA is a convenient tool to calculate and analyze data, and the research results are comprehensible. However, this period can only be regarded as the primary stage for SPA application in engineering geology, with only simple application and qualitative analysis.

In 2008, the development of SPA has entered a new era, as indicated by the sharp increase in the number of articles involving SPA application (Fig. 1). Engineering risk/safety evaluation is the main research field incorporating SPA during this period, such as safety assessment of a tailing pond (Chen and Lv 2012) and safety risk evaluation of an earth-rockfill dam (Zheng and Liu 2013). A few researchers have adopted SPA in the prediction and classification of engineering problems. For example, Bu et al. (2018) advanced a new SPA-based classification method and a tunnel seismic prediction system, which were proposed to accurately predict the surrounding rock classification. Some simple SPA-based models have been established in the above research fields to show the uncertain relations of the system, but the weight of each factor has not been discussed. Therefore, scholars attempted to use various mathematical methods to optimize the SPA theory. For example, the SPA model was improved by combination with the information entropy method (Wang et al. 2009a; Wang and Jin 2009). These authors presented a decision-making model by combining information entropy with SPA to discuss the weight of the uncertain information in detail, and then used it to select the best treatment scheme for landslide disaster assessment and foundation design. Based on the results, the optimized model can be more effective in decision-making (Wang et al. 2009a; Wang and Jin 2009).

Fig. 1
figure 1

Literature of SPA applied in different directions of geology (1997–2019)

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Optimization of SPA with various mathematical methods represents a new era for SPA development. On the one hand, the optimized SPA method was accepted and applied by more and more scholars, especially in decision-making, such as in product design (Wei et al. 2010) and priority determination for repairing earthquake-damaged transport infrastructures (Wang et al. 2016). Such research is characterized by the combination of deterministic decision models and uncertainty analysis, as well as by the integration of two or more decision-making methods. Thus, it is a kind of intelligent decision-making based on the overall situation. The optimized SPA method was also used in a study of surrounding rock stability evaluation (Qi and Zhang 2010), and different mathematical theories and methods were used to optimize the SPA theory to reduce the error in uncertainty analysis and to obtain more reasonable results. On the other hand, scholars have also attempted to improve the SPA theory. For instance, game theory (Liu et al. 2014) and rough sets (Jin et al. 2014; Huang et al. 2018) were generally used to determine the expression of the connection numbers, and analytic hierarchy process (Wu and Lin 2016), information weight (Wang and Li 2018), and the fuzzy set method (Li et al. 2019d) were used to determine the uncertainty coefficients. Furthermore, the optimized SPA method was widely employed in studies on engineering problems in this decade, which promoted the further development of SPA other than its simple application and qualitative analysis.

SPA Application and Development in Hydrology and Water Resources

Guo (1998) introduced the SPA method as a novel tool for water resource management and found that it could aptly reflect the uncertain factors in water resource management, such as the dynamic and uncertain hydraulic boundary. His research provided a new theoretical basis for the selection of water resource management schemes. Unfortunately, only a few scholars have acknowledged the advantages of SPA in dealing with uncertainty issues before 2006, and the few studies in this field focused on water quality assessment (Yu and Gong 1999; Wang and Wang 2002a; Li and Chen 2003) and surface water eutrophication evaluation (Li et al. 2000). The above-mentioned studies are derived from a set pair that combines actual water quality with each water quality evaluation criterion. In this sense, SPA was mainly applied to explore the uncertain relation among the factors influencing water quality evaluation systems. The application of SPA remained in the cognition phase, and this tool was merely used to show the uncertain relation of the factors in the evaluation system.

Wang et al. (2004) noticed that the uncertain correlation cannot be evaluated quantitatively, which is a major disadvantage of SPA. The authors then optimized it by using the fuzzy mathematical set theory. The improved model could better reflect the uncertain relations and connection degrees between water samples and evaluation criteria. In the same year, Feng et al. (2004) used SPA to statistically forecast the change tendency of water resources. Around this time, scholars began to realize that SPA can effectively deal with the uncertainty problems, and the identification, treatment, and application of various uncertainties in hydrological and water resources systems have been research hotspots in the past few decades. As a new attempt, SPA has been widely applied in studies of hydrology and water resources after 2006 (Fig. 2). Since then, it has been applied in water resources exploitation evaluation (Shao and Liu 2006), carrying capacity evaluation (Wan et al. 2006a; Zhang and Li 2010), water security evaluation (Yang et al. 2011a), and pollution evaluation (Feng and Sun 2006), mainly because of its strong capability of analyzing the relevance and transformation between the real situation and the evaluation criteria from different perspectives. However, SPA was mainly used in water resources assessment and was rarely applied in water resources analysis, decision-making, classification, and prediction. In addition, SPA could depict an uncertainty relationship among each index, but the uncertainty coefficients are generally assigned according to expert experience. Therefore, high subjectivity and randomness have become the fatal defects of SPA-based models, further limiting their applicability.

Fig. 2
figure 2

Literature of SPA applied in hydrology and water resource and the proportion of each research direction (1998–2019)

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Since 2010, scholars have continuously been optimizing SPA with numerous mathematical methods to improve its calculation accuracy and widen its applicability. Some new assessment models have been established, which can determine the uncertainty coefficients of the connection number in a more precise way to reduce the probability of error (Pan et al. 2009; Zhang et al. 2010; Li et al. 2011a). There are many different methods to determine the uncertainty coefficients, such as fuzzy mathematics (Gong et al. 2011), BP neural network (Wu et al. 2010), entropy weight (Zhang et al. 2016), confidence intervals (Mulenga and Li 2016), analytic hierarchy process (Zeng and Huang 2017), integrated k-means clustering (Li et al. 2016), and s-type function (Men and Liu 2018). They all have their advantages, making it necessary to analyze the practical problems to select the appropriate method. In addition, the application scope of SPA has gradually expanded, from previous evaluation research (Zhang et al. 2007) to risk analysis (Yang et al. 2011b; Li et al. 2011b; Yang 2016), decision-making (Dong et al. 2010), and management research (Yang et al. 2017; Cui et al. 2018; Roy and Datta 2019). In particular, the application of SPA in hydrological prediction was another research hotspot in this period. Scholars established various prediction models based on the concepts of connection number and partial connection number, and the core was to construct a quantitative relationship between the variables of the predicted objects and their influencing factors. Such studies include regional water demand prediction (Guo et al. 2009), runoff prediction (Ou et al. 2009), flood risk prediction (Li et al. 2009; Guo et al. 2014), precipitation prediction (Liu et al. 2011; Xiao et al. 2018), and sustainable regional water use prediction (Jin et al. 2012; Men et al. 2017). However, the application of SPA in water resources classification is still weak (Jin et al. 2009; Wu et al. 2016).

Overall, there are significant gaps between different research fields regarding the application of SPA in hydrology and water resources, and its prominent application field is still the evaluation domain (Wang et al. 2009b; Wu and Wang 2012; Yue et al. 2014). In addition, SPA is mainly used by Chinese researchers, and only a few articles have been published in international journals (Fig. 2). The reason is that Prof. Zhao, the founder of SPA theory, published the theory of SPA in Chinese rather than in English, limiting the application by foreign scholars.

SPA Application and Development in Meteorology, Climatology, and Atmospheric Environmental Science

The application of SPA in meteorology, climatology, and atmospheric environmental sciences can be generally divided into two fields: air quality evaluation and weather prediction (Fig. 3). Set pair analysis was first used by Li (1998) to evaluate air quality; the author expounded the basic idea of the SPA-based comprehensive evaluation method and presented the equations for calculating the coefficients in the comprehensive connection degree expression. This study provided a novel method of assessing air quality. After this, the SPA theory was further adopted to evaluate atmospheric quality regarding three individual pollutants: sulfur dioxide, nitrogen dioxide, and carbon monoxide (Wang and Wang 2002b); the results confirmed that the concept of the ternary connection number (Eq. 1) is simple and practical for atmospheric environmental quality assessment. Based on that, some atmospheric quality assessments were carried out in Xupu Bridge and in the Ma’anshan area (Zhang et al. 2003; Cao 2004), and further verified the applicability of SPA in atmospheric quality assessment through comparing the results of SPA with the results of traditional methods or by comparing the results of SPA with actual situations. In this period, the application of SPA in atmospheric studies was still in a simple application stage, and scholars just used the concept of the connection number (Eqs. 1 and 2) to evaluate air quality. Therefore, there are some shortcomings in the determination of the uncertainty coefficients of the connection number, affecting the accuracy of the research results. Cao (2004) pointed out that the application of this method needed to be further explored and studied. Up to 2017, Liu (2017) further improved the SPA method by introducing the variable weight theory to determine the coefficients of the connection number; this made the research results more scientific and reasonable.

Fig. 3
figure 3

Literature of SPA applied in meteorology, climatology, and atmospheric environmental science and the cumulative total of each research subject (1998–2019)

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The application of SPA in weather prediction studies can be traced back to 1999, when Wang (1999) advanced a new method to predict the weather based on the SPA theory and REEP (regression estimation of event probability); the author further compared the predicted results of the proposed method and of the probabilistic regression method to verify its applicability. He further indicated that the introduction of SPA improved the prediction accuracy. A precipitation prediction research (Feng 2000) followed closely, in which the author established a relationship between precipitation probability and each prediction factor by using SPA and repeatedly adjusted the boundary values of each predicted factor to make the results more appropriate for the actual situation. The studies above both provided a new perspective on the theory and practice of weather prediction. Since then, more and more scholars have been realizing the importance of SPA theory in dealing with the uncertainty problems of weather prediction. Moreover, SPA was also applied to predict extreme weather events such as sandstorms, rainstorms, and extreme temperatures (Chen et al. 2002; Wang and Guo 2006; Zhang 2005; Zhang et al. 2013). Due to the disadvantage of this SPA-based method, heavily relying on existing data, few climate change analyses and calculations were carried out (Fig. 3), and researchers optimized SPA by introducing other mathematical methods to explore the changing trends of climatic characteristics (Ma et al. 2011; Luo et al. 2009).

According to the above-mentioned review and the results reported previously (Fig. 3), there are numerous shortcomings in the application of SPA in meteorology, climatology, and atmospheric environmental science, reflected in two aspects. First, the studies based on the SPA method should be considered a preliminary stage, in part because the number of articles was relatively low, particularly the number of articles in English. Second, the development and application of SPA in meteorology, climatology, and atmospheric environmental science is uneven, and most researchers focus on prediction and evaluation studies.

SPA Application and Development in Ecology

The ecological environment is a complex system containing many uncertainty relationships among influencing factors. To deal with these uncertainty relationships, Li et al. (2001) introduced the SPA theory into ecosystem evaluation research and established an SPA-based model to evaluate the coordinated development among society, economy, and environment. With the help of the concept of the connection number (Eq. 2), the proposed model is simple and intuitive, because the number of indicators in the index system is not limited. Therefore, this study lays a good foundation for the application and development of SPA in ecology. Consequently, SPA was widely used in studies on ecological evaluation to solve various uncertainty problems. For example, Li and Chen (2002) introduced SPA theory into landscape ecology in the context of urban green area assessment to obtain a correlation among each assessment index. Similarly, Deng et al. (2006) established an uncertainty relationship between the indicators of ecological carrying capacity and the grading standards based on SPA. Also in 2006, Wan et al. (2006b) proposed an uncertainty relationship between the index for the division of eco-environment fragility and the evaluation criteria by using SPA. These studies confirm that the uncertain correlations among impact factors in the whole system could be solved by SPA.

Based on previous studies, Su et al. (2007, 2009) adopted the SPA-based method to evaluate urban ecosystem health, indicating that the application of SPA not only avoids assigning weights empirically, but also makes the evaluation system no longer subject to the limitation of spatial and temporal dimensions. In 2016, Wang and Zhou (2016) reported that the application of SPA still has shortcomings in parameter determination, and the authors optimized the SPA theory by a growth curve function to obtain a comprehensive assessment method for the coordination ability of sustainable social–ecological system development; this method facilitated the determination of parameters by the growth curve function. Since then, scholars have adopted numbers of mathematical methods to improve SPA, with the aim to further determinate the parameters of SPA. Generally, the fuzzy approach, the relative membership degree, the cloud model, and the Markov chain model are the mathematical methods frequently used to improve SPA (Xue et al. 2010; Yan et al. 2016; Yan and Xu 2017; Zhao et al. 2018), making the results more reasonable and accurate. However, the results differ for different optimization methods, requiring further studies.

According to the results of literature statistics and analysis results (Fig. 4), numerous studies have focused on evaluation research, while only few studies involved ecological calculations and predictions (Liu 2011). The development progress of SPA in ecology is slow compared with its application in hydrology and water resources. Moreover, the theory of SPA has not been further developed in ecological studies, mainly because it was simply applied in ecological evaluation studies after optimization.

Fig. 4
figure 4

Literature of SPA applied in ecology and the proportion of each research direction every year (2001–2019)

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SPA Application and Development in Agriculture

In 2000, uncertainty problems in the study of different plant community characters started to be solved by SPA. For this, Du and Du (2000) used the concept of the connection number to prove that the characteristics of a whole plant community are not necessarily equal to the sum of the characteristics of each plant in the community. Since then, the SPA theory has been applied in agricultural studies. Shortly afterwards, Huang et al. (2001) established a SPA-based decision-making method to select associated tree species, analyzing the similarity between the ideal plan and the alternatives through SPA. These studies both established an uncertainty relationship among different factors in the system based on the connection number to obtain more reasonable results. However, the weights of each factor and the parameters of the connection numbers have not been explored in detail. In 2006, Qiu et al. (2006) optimized SPA to obtain a more reasonable method for synthetical bamboo species selection, and pointed out a new type of relative similarity degree based on the SPA theory, with the aim to obtain the ecological adaptability evaluation results of different mixed bamboo species and to avoid the weight selection of each evaluation index.

After this, scholars adopted several improved SPA methods to address different issues in agricultural research. For example, Yang and Lou (2008) used the regression calculation method to optimize the SPA theory and indicated a certain calculation formula of the parameters in the connection numbers to establish the relationship between crop yield and main climatic factors. Similarly, to determinate the parameters of the connection number, the concept of the maximum average connection degree was proposed by Tian et al. (2009), who applied it in the assessment of farmland use intensity. The above-mentioned studies explored the parameter determination in different ways. While some new concepts were proposed, it was shown that SPA was preliminarily developed for the field of agriculture. However, another shortcoming of the application of SPA in agriculture was pointed out by Shan and Lai (2011), who indicated that precision in data processing was insufficient when they applied SPA to analyze and calculate the security situation of cultivated land resources. Subsequently, various researchers tried to improve its accuracy by combining it with various data-processing methods. For example, GIS was used to deal with geographic information data (Tao et al. 2014), the fuzzy set method was used to process incomplete data (Cui et al. 2014), and the extended Fourier amplitude sensitivity test was used to sort out the data with poor regularity (Luan et al. 2018).

Based on this, and on Fig. 5, it can be stated that the application of SPA in agriculture is still at a very early stage, in part because of the scarcity of articles, but also because there was no clear definition of the new concept when it was proposed to improve the SPA.

Fig. 5
figure 5

Literature of SPA applied in agriculture and the proportion of each research direction (2000–2019)

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Challenges and Future Prospects of SPA in Earth and Environmental Sciences

Our literature review indicates that SPA has undergone rapid development in earth and environmental sciences due to the advantages of a simple concept, convenient calculation, and reasonable results (Chen et al. 2007). One of the indicators of this rapid development is the increasing number of studies in the past 20 years, as shown in the bibliometric analysis results (Fig. 6). In particular, the number of publications in international journals has been increasing in the past 10 years. However, as a new uncertainty analysis theory, the theoretical basis of SPA is still weak, and there is no complete theoretical and methodological system (Jin et al. 2019). This can result in considerable challenges in the application and development of SPA in the earth and environmental sciences, and the following issues require further investigation.

Fig. 6
figure 6

Literature of SPA applied in earth and environmental science and the article counts of each subject every 5 years (1997–2019)

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First, the connection number is the core concept of SPA, but its application is linked with two difficulties. One is the determination of the weight of each index, and the other is the calculation of the connection number parameters. To solve these problems, various mathematical methods have been used to improve the SPA theory. However, the calculation results from different methods are slightly different, impeding decision-making. Thus, it is necessary to explore a unified scientific and reasonable method to calculate the parameters of the connection number and to make the expression of the connection number more scientific (Yang 2018; Jin et al. 2017).

Second, several new concepts have been put forward to optimize the SPA theory, especially in prediction and decision-making studies. However, there is no complete unified system for these new concepts (Liu et al. 2008), and therefore, these new concepts have been used with their own subjective assumptions without clear rules. It is therefore important to provide a clear definition of a new concept rather than a calculation formula.

Third, the application and development of SPA in earth and environmental sciences are extremely unbalanced, and in hydrology and water resources SPA is applied and developed frequently and rapidly, mainly in the evaluation and prediction of water resources. This will negatively influence the development of SPA in other research fields in the long run, making it necessary to enhance the application of SPA in other research fields, especially in meteorology, climatology, and atmospheric environmental science.

SPA has achieved some noticeable advances in application in earth and environment sciences during the past 30 years in dealing with uncertainty problems in many disciplines of earth and environment sciences. It can be foreseen that the development and application of SPA will be broadened in the future, as it is being accepted by more and more researchers as a strong tool to deal with uncertainty problems. However, most of the users currently are Chinese scholars and promoting it to the international community is still a big challenge. Here we give some suggestions to further development of SPA. (1) Organizing some national and international symposia is necessary and helpful to introduce this concept to the international community. The symposium can be held regularly, e.g., making it an annual or biannual event. (2) Delivering some training courses is also helpful for the propagation of this theory. Considering the cost of the training courses, it is better that students can enjoy some discounts. (3) Encouraging transdisciplinary application of the SPA theory, and publishing high quality research of SPA in international journals that have broadened international readers are useful to promote the concept of SPA.

Concluding Summary

The research and application fields of SPA have been expanded continuously, covering almost all research fields of earth and environmental sciences. The SPA theory is a significant innovation with high academic importance and practical value, because it enriches and improves the traditional methods for analysis, calculation, prediction, evaluation, and decision-making. However, its application is still linked with serious issues, such as imbalanced development in each discipline and an immature concept system. Thus, exploring a unified method that is more scientific and reasonable for calculating the parameter (i and j) and constructing a scientific concept system may be the research hotspots in the future development of SPA.