The business of Hydrology is to solve the water balance equation (Dooge, 1986). Quantitatively, the hydrological perspective towards the study of the Earth's water is reflected in the water balance equation. Hydrology was defined as “the science that seeks to explain and quantify the water balance dynamics for any defined spatial scale (from a point to global) and temporal scale (from seconds to years) and their relationships with the physical and chemical transport of matter through the hydrological cycle and with ecology” (Lee, 1992). The role of the hydrologist is to integrate the findings of the other allied sciences to explain the dynamics of water balance over an area in any defined time period and to establish their relationships to the physical, chemical and biological environments. Those three italicised words in the previous sentence could be taken into disciplinary consideration for the evolutionary stages of the hydrological sciences.

In this paper, according to the descriptive-explanative-humanistic ideology, the authors summarise the evolution routes and development stages of hydrological sciences from the dimensions of research objective (or problem), discipline and methodology, respectively. Some thoughts or ideas will be provided for future directions of Hydrology, especially at the intersection of the evolution routes in terms of research objective, scientific recognition, and technical innovation in the field of hydrology. It seems to us that the 3-dimensional diagram of evolution routes of hydrological sciences gives support to Panta Rhei, the new IAHS Scientific Decade 2013–2022 (Montanari et al., 2013), focusing on “Change in Hydrology and Society” for hydrological research to cope with the changing human and environmental systems in the real world.

There exists many approaches to solving hydrological issues, such as steady and unsteady solutions with respect to time, homogeneous and heterogeneous methods with respect to space, saturated flow and unsaturated flow, continuous and discrete means, analog and digital methods, analytic and numerical approaches, linear and nonlinear methods, deterministic and stochastic approaches, white-box and black-box methods, physically-based and conceptual approaches, lumped and distributed methods, empirical methods and theoretical formulas. Due to the very significant variations in systems, including atmospheric, geochemical, hydrological and ecological processes, and the ceaseless interaction among hydrosphere, atmosphere, lithosphere, and biosphere, the collection, processing and analysis of water-related information in such a complex system should be established on the basis of Geosciences. During the 24th IUGG Assembly at the University of Perugia, Italy, July 2007, the IAHS PUB Scientific Decade 2003–2012 resulted in the suggestion that hydrologists should move from fitting hydrological data to fitting catchment characteristics.

Generally speaking, an approach to hydrology, different from classical mathematical and physical methods (such as mechanics, thermotics or thermodynamics), was regarded as a simplified approach (such as Horton infiltration formula, rainfall–runoff correlation, unit hydrograph, Muskingum routing, rational formula). Although mathematical and physical methods were restricted in the application of hydrology because of complex initial and boundary circumstances, they would still be the criteria to judge whether an approach to hydrology is correct or not. For instance, the Saint-Venant equations are the theoretical basis of channel flood routing approaches. The routing approach is right (wrong) if (not) in accordance with the equations (Zhao, 1994).

The evolution of approaches of hydrology might be summarised as ranging from empirical approaches to theoretical ones. At the beginning, hydrologists were not in a position to completely understand the theoretical basis of an approach to hydrology, but it might be found later on. The approach to hydrology was empirical at the beginning and the physical nature of formulae or the physical meaning of parameters was understood later on. For example, the Horton infiltration formula would be an approximate solution of the Richards equation under specific conditions, and the unit hydrograph could be derived theoretically from linear system theory, such as the Nash-form IUH.

Dooge (1988) pointed out that progress in the development of hydrological theory has been hampered by a fragmentation of the approach to the subject and by a failure of communication between those using different techniques. The deterministic approach and the stochastic one have too often been considered as rivals to one another rather than as complementary. In fact the two approaches have similar response functions (Dooge, 1988). Ren (1992) proposed the compatibility between two approaches in his thesis at University College Galway by comparing Bernoulli and Poisson processes in terms of probability distributions, specifically for the Nash-form IUH case (Table 1). It is recognised that deterministic and stochastic approaches have been combined together for hydrological forecasting (such as linear perturbation models, probabilistic forecasting of hydrological variables) and for the quantitative manifestation of hydrological uncertainties since the 1980s. Grayson and Bloeschl (2000) gave a typical example showing that a combined deterministic (by soil type) and stochastic (by soil hydraulic conductivity) pattern produced patterns of runoff occurrence that were most similar to the observed patterns in the tropical environment.

It should be emphasised that hydrological experiments have been of great significance with respect to the development of hydrological disciplines, e.g. Darcy's law and Dalton's formula were derived from experiment and observation. It is true that progress in the field of hydrology, including steps from empiricism to conceptualisation and scientificity, and from qualitative description to quantitative explanation, has much to do with experimental measurement. As a result of a very complicated water system under an ever changing environment, it makes sense to conduct experimental research in sensitive regions to understand the hydrological cycle, matter and energy equilibria and to recognise the eco-hydrological response mechanisms in a changing environment, and to estimate ecological flow for environmental demand in rivers covering various climatological and geographical zones.

The development of science (or a scientific discipline) may be attributed to two driving forces. One is the curiosity of scientists, and the other is the demand from society. The development of hydrology is no exception. In different periods of social development, hydrologists have undertaken the task of solving water-related problems for the national economy and people's livelihood. Meanwhile the discipline of hydrology has advanced by means of theoretical application and practical verification. On the basis of the descriptive-explanatory-humanistic ideological system, three main routes of evolution in hydrological sciences may be summarised as in the following sections.

Climate change and increasing population, two driving forces in the dynamic earth at present, are converging to create a water crisis affecting people and the environment. Eighty-three per cent of the land surface of this planet has been modified by the engineering activities of humans, in most cases without any understanding of how the impacts of these modifications have changed processes in natural ecosystems, such as biogeochemical cycles and energy flows. Virtually all these modifications have had direct and indirect impacts upon disturbances of the water cycles, sometimes both locally and globally (Harper et al., 2008).

Landscapes have changed significantly in China in the last 20 years and the amount and rate of change have been greater here than elsewhere in the world due to the rapid economic growth in this period. It can be seen from the typical semi-arid example of the Laohahe basin of northern China (Liu et al., 2009; Yong et al., 2013) that human activities have played a dominant role in streamflow alteration. Predicting the response of hydrologic systems in a changing environment requires that the water cycle be considered as a complex, interconnected system that includes not just the physical but also the biogeochemical, ecological, and human subsystems whose interactions contribute to land-forming (i.e. structure-forming) and life-sustaining processes (Wagener et al., 2010). Because of such a need, IAHS launched the New Scientific Decade 2013–2022: Panta Rhei during the IAHS-IAPSO-IASPEI Joint Scientific Assembly in Gothenberg in July 2013. It contains three targets, viz. understanding, prediction and estimation; science in practice; and six science questions, so as to cope with the changing environmental system (Montanari et al., 2013).

We expect that the three-dimensional diagram of the evolution routes and stages of hydrological sciences, as shown in Fig. 1, would provide some thinking and support for dealing with those science questions of the Panta Rhei decade, particularly in the aspects of methodology and discipline. For example, based upon a platform of digital basin, the means of digital hydrology can help answer the 4th science question “How can we use improved knowledge of coupled hydrological-social systems to improve model predictions, including estimation of predictive uncertainty and assessment of predictability?”, since such a coupled system definitely contains a huge amount of information.

In addition, the intersection among different stages along the main routes will help effectively link climatic, hydrological, and agricultural models with social and economic analyses. Hydrological processes include both deterministic and indeterministic components due to their variability in space and time, and diversity in pathways of the water cycle. Accordingly, the approach to hydrological modeling could be a combination of deterministic and stochastic methods, the organisation of deductive and inductive approaches. In this aspect, a good example was given for hydrologic synthesis across processes, places, and scales (Bloeschl, 2006) in terms of runoff prediction in ungauged basins (Bloeschl et al., 2013), as the milestone outcome of the past IAHS PUB Decade 2003–2012.

Processes and elements of the hydrological cycle have changed due to changes in both upper boundary (climate change) and lower boundary conditions (land-use/cover, population growth, water supply increasing, socioeconomic development). It may result in water-related disasters as well as hydrological risk (Ren et al., 2012). The desire for people to intrinsically integrate natural, economic and social processes (dynamic) over multiple scales of space and time is becoming stronger and stronger. One would like to know, where the problem is more serious and what factors cause these problems in the real world. For instance, what is the relationship between eutrophication and land-use change? What are the best measures for mitigation? What types of crops should be planted within a catchment, so as to achieve both a maximum benefit in agriculture and to exert a minimal impact on water quality at the same time? To what extent might ecological property be deteriorated in the organised case of hydrological regimes and water quality which have changed? Answers to these questions requires a new approach such as eco-hydrology to balance water for humans and nature (Falkenmark and Rockström, 2004), and to check or review human socioeconomic activities. Additionally, digital hydrology may be capable of providing a new platform for a distributed framework and information integration (Liu, 2000).