Briefly tracing the history of hydrologic modeling, this paper discusses the progress that has been achieved in hydro - logic modeling since the advent of computer and what the future may have in store for hydrologic modeling. Hydro- logic progress can be described through the developments in data collection and processing, concepts and theories, integration with allied sciences, computational and analysis tools, and models and model results. It is argued that with the aid of new information gathering and computational tools, hydrology will witness greater integration with both technical and non-technical areas and increasing applications of information technology tools. Furthermore, hydrol- ogy will play an increasingly important role in meeting grand challenges of the twenty-first century, such as food security, water security, energy security, health security, ecosystem security, and sustainable development. Keywords: Hydrologic models, Data processing, Computational tools, Hydrologic advances, Future outlook 1960s, many groundbreaking advances in modeling dif- Introduction ferent components of the hydrologic cycle were made. Hydrology has a long history dating back to several mil- Some of these advances were based on the laws of math- lennia (Biswas 1970). However, the birth of hydrologic ematical physics and some had their basis in laboratory modeling can be traced to the 1850s when Mulvany and/or field experiments. The current state of hydrologic (1850) developed a method for computing the time of science and engineering owes a great deal to the pre-1960 concentration and hence the rational method for com- advances. The handbook of applied hydrology edited by puting peak discharge which is still used for urban Chow (1964) provided an up-to-date account of hydro- drainage design, Darcy (1856) who conducted experi- logic advances until the 1960s, whereas the handbook of ments on flow-through sands and developed what is now hydrology edited by Maidment (1993) and the encyclope- referred to as Darcy’s law which laid the foundation of dia of hydrology and water resources edited by Hershey quantitative groundwater hydrology, and Fick’s first law and Fairbridge (1998) dealt with advances that occurred which states that under steady-state conditions the dif- during the intervening period. Singh and Woolhiser fusive flux is proportional to the concentration gradi - (2002) provided a historical account of developments ent (spatial) which laid the foundation of water quality that occurred in modeling different components of the hydrology. About half a century earlier, Dalton (1802) hydrologic cycle. formulated the law of evaporation which states that the The decade of the 1960s witnessed the birth of com - rate of evaporation is directly proportional to the dif- puter revolution and hydrologic modeling took a giant ference between saturation vapor pressure at the water leap forward. The computer provided the power for surface and the actual vapor pressure in the air. This law doing computations that was not available before. As constituted the foundation for developing the physics a result, a new branch of hydrology, called digital or of evaporation. For a period of over a century until the numerical hydrology, was born. Another branch that came into being was statistical or stochastic hydrology that often required analyses of large volumes of data. *Correspondence: email@example.com Department of Biological and Agricultural Engineering & Zachry Then, several major advances ensued. First, simula - Department of Civil Engineering, Texas A&M University, 321 Scoates Hall, tion of the entire hydrologic cycle became a reality, as 2117 TAMU, College Station, TX 77843-2117, USA © The Author(s) 2018. This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creat iveco mmons .org/licen ses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made. Singh Geosci. Lett. (2018) 5:15 Page 2 of 18 illustrated by the development of the Stanford Watershed came into being that made possible to acquire spatial Model (Crawford and Linsley 1966) which was followed data for large areas (Engman and Gurney 1991; Hogg in the decades to come by umpteen watershed mod- et al. 2017; Lakshmi 2017; Lakshmi et al. 2015). Likewise, els that were developed all over the world (Singh 1995; geographical information systems (GIS) were developed Singh and Frevert 2002a, b, 2006). Second, optimiza- for processing huge quantities of raster and vector data tion or operations research techniques were developed, (Maidment 2002). The past two decades witnessed the which formed the basis for reservoir management and development of artificial neural networks, fuzzy logic, operation as well as river basin simulation. Some of these genetic programming, and wavelet models (Kumar et al. techniques were also used for calibrating hydrologic 2006; Ross 2010; Sen 2010; Tayfur 2012). New theories models (Beven 2001; Duan et al. 2003). Third, two- and borrowed from other areas were introduced in hydrol- three-dimensional modeling was made possible because ogy. Examples of these theories are entropy theory (Singh of advances in numerical mathematics. Consequently, 2013, 2014, 2015, 2016, 2017b), copula theory (Singh and two- and three-dimensional models of groundwater as Zhang 2018), chaos theory (Sivakumar 2017), network well as of infiltration and soil water flow were developed theory (Sivakumar et al. 2017), and catastrophe theory (Bear 1979; Pinder and Celia 2006; Remson et al. 1971). (Poston and Stewart 1978; Zeeman 1978). These theories Fourth, simultaneous simulation of water flow and sedi - will find increasing place in hydrologic modeling in the ment and pollutant transport was undertaken; likewise, years ahead. simultaneous simulation of different phases of flow, such Another area that mushroomed subsequent to the pre- as liquid and gaseous, was done (Bear and Verruijt 1987; computer era is instrumentation. New instruments which Charbeneau 2000). Fifth, modeling at large spatial scales, were more accurate and sophisticated were developed such as a large river basin like the Mississippi, and that at for measuring all kinds of hydrologic variables, such as small temporal scales, such as seconds or minutes, was velocity, soil moisture, water and air quality parameters, undertaken (Molley and Wesse 2009; Sorooshian et al. fluxes in porous media, energy fluxes, and so on. Fur - 2008). Sixth, integration of hydrology with allied sciences ther, instrumentation for data transmission from place became possible. For example, it was possible to couple of measurement to place of storage, processing, storage, hydrology with climatology for precipitation modeling retrieval, and dissemination became highly robust and and forecasting (Sorooshian et al. 2008), with geomor- accessible (Liang et al. 2013; Sivakumar and Berndtsson phology for river basin geometric representation (Baker 2010). et al. 1988; Bates and Lane 2002; Beven and Kirkby 1993), The objective of this paper, therefore, is to provide a with hydraulics for describing flow characteristics (Singh snapshot of major advances that have occurred for over a 1996), with soil physics for quantifying soil texture and century and a half, discuss where hydrology is headed as structure (Bohne 2005; Guymon 1994; Miyazaki 2006; a science and engineering, and conclude with a personal Smith et al. 2002; Singh 1997), and with geology for aqui- reflection on future outlook. fer characterization (Delleur 1999; Fetter 1980; Singh 2017a, b, c). The coupling of hydrology with ecosystems History of hydrologic developments gave rise to ecohydrology (Eagleson 2002; Gordon et al. There have been a large number of developments in 2006; Rodriguez-Iturbe and Porporato 2004). Climate hydrology since the 1850s, so it will be difficult to do jus - change and global warming became part of hydrologic tice to describe all of them. Therefore, only a snapshot of analysis (Arnell 1997). A more detailed account of devel- some of the major developments from a personal per- opments in different components of the hydrologic cycle spective will be provided. For convenience of easy refer- is given in Singh (2013, 2014, 2015, 2017a). ence, these developments will be organized topic-wise In the decades that followed, computing prowess rather than chronologically. increased exponentially and hydrology began maturing and expanding in both depth (vertically) and breadth Watershed geomorphology (horizontally). Tools from fluid mechanics, statistics, In 1945 Horton, derived a set of empirical laws that are information theory, and mathematics were employed and now called Horton laws which laid the foundation of became part of hydrology (Bras and Rodriguez-Iturbe quantitative geomorphology. These laws were the law 1985; Clarke 1998; Gelhar 1993; Mays and Tung 1992; of channel numbers, law of channel lengths, and law of Singh et al. 2007; Tung and Yen 2005). Further, computer stream slopes. He developed a scheme for channel and also made possible the development of user-friendly soft- basin ordering, called Horton ordering. Horton (1932) ware, and tools for date acquisition, storage, retrieval, also defined drainage density and length of overland processing, and dissemination (Croley 1980; Hoggan flow. He investigated landform development and stream - 1989). Remote sensing tools, such as radar and satellites, flow generation dominated by overland flow. Strahler Singh Geosci. Lett. (2018) 5:15 Page 3 of 18 (1952) modified Horton’s method for ordering channel 1980), maximum sediment discharge theory (White et al. networks which is now referred to as Horton–Strahler 1982), maximum friction theory (Davies and Sutherland ordering scheme. Schumm (1956) developed the law of 1983), minimum unit stream power theory (Yang and stream areas. Because discharge is highly correlated with Song 1986), thermodynamic theory (Yalin and da Silva drainage area, as shown by Hack (1957) for mean annual 1997, 1999), minimum energy dissipation theory (Rodri- discharge and Leopold and Miller (1956), and Gray and guez-Iturbe et al. 1992), principle of least action (Huang Wigham (1970), a law of discharge can be formulated and Nanson 2000), and entropy theory (Deng and Zhang as shown by Singh (1992). Strahler (1957) formulated 1964; Singh et al. 2003a, b; Singh and Zhang 2008a, b). the law of drainage basin similarity, but Gray (1961) Each theory leads to unique hydraulic geometry rela- showed that not all basins possessed geometric similar- tions, meaning different values of exponents. Singh ity. Gray (1961) established the relation between drain- (2003) has discussed characteristics of these relations age area and length which was also investigated by Smart with regard to their basis, tendency to equilibrium state, and Surkan (1967). Shreve (1966) developed a statistical limitations of the equilibrium assumption, validity of law of channel numbers. Using the theory of minimum power relations, stability of exponents in power relations, energy dissipation rate, Yang (1971) developed the law of effect of channel patterns, effect of stream size, depend - average stream fall. Much of the progress made in sub- ence of exponents on climatic and environmental factors sequent years draws heavily from these foundational con- and land use, extension to drainage basins, and impact of tributions. Fitzpatrick (2017) has reported on watershed boundary conditions. geomorphologic characteristics. Smith (1974) derived hydraulic geometry of steady state channels from con- Surface runoff servation principles and sediment transport laws. Using In 1850, Mulvany developed a method, called rational entropy theory and theory of minimum energy dissipa- method, for computing peak discharge due to a rain- tion rate, Singh et al. (2003a, b) derived a hierarchy of fall event with uniform intensity and duration equal to downstream hydraulic geometry and Singh and Zhang or greater than the time of concentration. The method (2008a, b) upstream hydraulic geometry. Applications was meant for small urban watersheds which are in use of channel network are included in Beven and Kirkby for urban drainage design to date. St. Venant de (1871) (1993) and flood geomorphology is presented in Baker derived equations for modeling surface flow and these et al. (1988). Rodriguez-Iturbe and Rinaldo (2001) have equations are now called St. Venant equations. Two described river basins using fractal geometry. The water - decades later, Manning (1895) developed an equation shed geomorphology has played a fundamental role in for computing flow velocity in open channels. Imbeau developing runoff prediction models for ungauged basins (1892) developed a relation between storm runoff peak (Bloschl et al. 2013; Wagner et al. 2004). and rainfall intensity. Sherman (1932) developed the unit hydrograph concept which laid the foundation of linear Hydraulic geometry systems hydrology. Horton (1939) derived a semi-empir- Hydraulic geometry is of two types, at-a-station and ical formula for overland flow. Barnes (1940) developed downstream, and encompasses relations of channel a technique for hydrograph separation. Applying hydrau- width, depth, velocity, roughness, and slope each with lic principles, Keulegan (1944) showed the adequacy of discharge (Wolman 1955). Leopold and Maddock (1953) simplified momentum equation for modeling overland derived these hydraulic geometry relations which are flow. Izzard (1944) conducted experiments on over - of power form. Because of their great practical value land flow on paved surfaces. Clark (1945) developed a in design of stable channels, river flow control works, unit hydrograph method for deriving the rainfall–run- river improvement works, and irrigation schemes, there off hydrograph. These contributions laid the foundation is a large body of literature describing the derivation of for conceptual as well as physically based rainfall–runoff these relations using different types of theories, including modeling. However, for application of these methods the regime theory (Blench 1952), tractive force theory (Lane amount of surface runoff was assumed to be known and, 1955), minimum entropy production theory (Leopold therefore, rainfall excess was known. and Langbein 1962), stability theory (Stebbings 1963), In 1956, the Soil Conservation Service (SCS) [now minimum variance theory (Langbein 1964), minimum called National Resources Conservation Service (NRCS)] channel mobility theory (Dou 1964), minimum energy of the U.S. Department of Agriculture (USDA) devel- degradation theory (Brebner and Wilson 1967), threshold oped a method, now called SCS-Curve Number (CN) geometry theory (Li 1974), hydrodynamic theory (1974), method, based on a large amount of data, for computing minimum stream power theory (Chang 1980), maximum the amount of runoff generated by a rainfall event, tak - sediment discharge and Froude number theory (Ramette ing into account abstractions, antecedent soil moisture Singh Geosci. Lett. (2018) 5:15 Page 4 of 18 condition, hydrologic condition of land use and land showed that diffusive and kinematic wave approxima - cover, and soil type through curve number. This method tions would suffice for most cases. Singh (2017a , b, c) is still quite popular for determining the amount of run- presented the kinematic wave theory of surface runoff. off or rainfall excess from small and medium agricultural Woolhiser and Liggett (1967) derived the kinematic wave watersheds, and has been extended to urban and forested number which served as a criterion for the kinematic watersheds. Nielsen et al. (1959) investigated the source- wave approximation. This work gave the real impetus to area contribution to runoff. the popularity of kinematic wave approximation. Morris In 1956, the U.S. Army Corps of Engineers published and Woolhiser (1980) revised the kinematic wave num- the summary report of the snow investigations as a ber with the use of Froude number. Singh (1994) derived book entitled “Snow Hydrology” that laid the founda- the error differential equation for judging the accuracy tion for much of the work that has since ensued. The of kinematic and diffusive approximations. Moramarco book described virtually all aspects of the snow environ- et al. (2008a, b) made a comprehensive analysis of the ment. Martinec (1960) developed a degree-day method accuracy of kinematic wave and diffusion wave approxi - for determining snowmelt. Anderson (1968) developed mations. Kibler and Woolhiser (1972) developed the and tested snowpack energy balance equations. Colbeck kinematic cascade. Smith and Woolhiser (1971) explic- (1972) developed a theory of water percolation in snow itly incorporated infiltration in overland flow modeling. and Colbeck (1975) developed a theory of water move- Berod et al. (1999) developed a geomorphologic kine- ment through a layered snowpack. Gray and Prowse matic wave model. These investigations established that (1993) provided an excellent discussion of different the kinematic wave approximation would be sufficiently aspects of snow and floating ice. Singh et al. (1997a , b) accurate for surface runoff modeling and has since been a developed the kinematic wave theory of vertical move- standard technique. Singh (1996) prepared two treatises ment of snowmelt water through snowpack and of satu- on kinematic wave modeling in surface water hydrology rated basal flow in a snowpack. Kuchment (2017) proved and environmental hydrology that comprehensively sum- an excellent review of snowmelt runoff generation and marize the kinematic wave literature. modeling. Singh et al. (2011) prepared an encyclopedia of snow, ice and glaciers. Reservoir and channel flow routing Nash (1957) developed a theory of instantaneous unit Analogous to surface runoff modeling, both hydrologic hydrograph (IUH) that led to what is now called the systems and physically based techniques have been Nash model. Nash (1959) also developed the theory of applied to route flows through reservoirs and channels. moments for determining his model parameters. Dooge Puls (1928) presented a method for reservoir flow rout - (1959) developed the generalized unit hydrograph the- ing. MeCarthy and others (U.S. Army Corps of Engineers ory that included the Nash IUH theory as a special case. 1936) developed the Muskingum method for routing These IUH theories led to the development of systems of flow in channels. Kalinin and Miljukov (1957) devel - hydrology detailed by Singh (1988, 1989) in which sys- oped a unit hydrograph model for channel flow rout - tems techniques can be applied to flow routing, base ing. Cunge (1969) developed a method for estimating flow, water quality routing, erosion and sediment trans - the Muskingum method parameters from hydraulic and port. Combining laws of geomorphology with the IUH channel geometry characteristics. Since then, the Musk- theory Rodriguez-Iturbe and Valdes (1979) developed the ingum method has been a popular method and its several geomorphologic unit hydrograph that has since received variants have been developed. Koussis (2009) provided a great deal of attention and is now frequently used in an assessment and a review of the hydraulics of storage practice. flood routing 70 years after the introduction of the Musk - Physically based surface runoff modeling was based ingum method. on the St. Venant equations and simplifications thereof Stoker (1953) and Isaacson et al. (1954, 1956) used the whose solutions required the use of numerical algorithms complete St. Venant equations for flood routing in the and became popular in the 1960s and the ensuing dec- Ohio River. Abbott (1976) and Grupert (1976) summa- ades. Depending on the simplification, these equations rized the flood routing models. Fread (1984) developed give rise to five types of waves: dynamic waves, steady a one-dimensional dynamic wave model in a single or dynamic waves, gravity waves, diffusive waves, and kin - branched waterway. Linear forms of the St. Venant equa- ematic waves, and hence five types of models. Using dif - tions were employed since the work of Dooge (1967), and ferent techniques, Lighthill and Whitham (1955), Iwagaki Dooge and Harley (1967). Kundzewicz (1986) discussed (1955), Woolhiser and Liggett (1967), Ponce and Simons physically based flow routing methods. Abbott (1979) (1977), Menendez and Norscini (1982), and Ferrick presented numerical methods for solving free surface (1985) analyzed the characteristics of these waves. They flow equations. Singh Geosci. Lett. (2018) 5:15 Page 5 of 18 Hayami (1951) employed diffusion wave approximation to as the combination method for computing evaporation for flood routing. Lighthill and Whitham (1955) showed from saturated water bodies as well as vegetated surfaces. that diffusion waves were described by a convection–dif - The Penman method laid the foundation for subsequent fusion equation. Cunge (1969) showed the connection developments in the evaporation field. Budyko (1955, between Muskingum method and convection–diffusion 1974) prepared an atlas of heat balance of Earth. Mon- equation. Huang (1978) used a finite difference solution teith (1965, 1973, 1981) modified the Penman method of kinematic wave equation for routing flows in channels. which is now called the Penman–Monteith method. Singh (1996) has given a full account of different routing Morton (1965, 1969) developed a method, called com- methods. Perumal and Price (2017) have reviewed reser- plementary method, for computing regional evapora- voir and channel routing. tion. Priestley and Taylor (1972) developed an equation for computing evaporation. Doorenbos and Pruitt (1977) Interception and depression storage developed methods for computing evapotranspiration Interception loss in humid forested watersheds may and hence crop water requirements. Jensen and Allen account for as much as 25% of annual precipitation. (2016) have comprehensively summarized methods for Helvey and Patrick (1965) found that this loss might be computing evaporation, evapotranspiration, and irriga- of the order of 15 cm for such watersheds. Horton (1919) tion water requirements. Hobbins and Huntington (2017) developed a series of empirical equations for computing have provided an up-to-date account of evapotranspira- storm interception for a variety of vegetative covers. Lin- tion and evaporative demand. sley et al. (1949) developed an exponential type model for computing interception by vegetation. Merriam (1960) Infiltration and soil water flow modified the Horton model. Bultot et al. (1972) derived Infiltration is fundamental for computing surface runoff empirical relationships for computing interception loss. modeling, groundwater recharge, and agricultural irriga- Deguchi et al. (2006) computed the influence of seasonal tion. In 1911, using physical principles Green and Ampt changes in canopy structure on infiltration loss. Gash developed a formula for computing infiltration capac - (1979) developed an analytical model for infiltration loss ity rate which is one of the most commonly used infil - by forests. Gerrits et al. (2010) discussed the spatial and tration formulae today. Richards (1931) derived what is temporal variability of canopy and forest floor intercep - now called Richards equation for modeling flow-through tion in a beech forest. unsaturated soils (Richards 1931, 1965). This equation Horton (1939) and Holtan (1945) empirically evalu- laid the foundation for vadose zone hydrology. Kostia- ated depression storage. Turner (1967) derived curves kov (1932) derived an empirical equation for computing for depression storage intensity as a function of time for infiltration capacity rate. Horton (1933, 1939) developed different antecedent conditions. Using a digital surface a theory of infiltration which was based on a hydrologic model, Ullah and Dickinson (1979a, b) investigated geo- systems concept. Horton (1940) tested his infiltration metric properties of depressions for hydrologic modeling. theory on experimental plots. Philip (1957) developed a Soil Conservation Service (1956) included interception theory of infiltration that led to Philip infiltration equa - and depression storage losses as a fraction of maximum tion. Mein and Larson (1973) developed a model for soil moisture retention capacity in the SCS-CN model computing infiltration under steady rain. Fok (1987) sum - (Mishra and Singh 2010c). Linsley et al. (1949) presented marized developments in infiltration and its application. an exponential model for computing surface depression Singh and Yu (1990) developed a generalized framework storage for a given effective rainfall. Borselli and Torri for infiltration and derived several popular infiltration (2010) discussed the relationship between surface storage models as special cases. Smith et al. (2002) prepared a and soil roughness and slope on impervious areas and treatise on infiltration theory for hydrologic applications. suggested an empirical model. Corradini et al. (2017) have reviewed the state of art of infiltration modeling. Evaporation Evaporation and evapotranspiration are amongst the Subsurface flow most important components of the hydrologic cycle and Subsurface flow is also referred to interflow and is some - their significance increases with the increase in timescale. times divided into quick interflow and delayed interflow Richardson (1931) and Cummings (1935) investigated (Chow 1964) and generates subsurface runoff. Low - evaporation from lakes. Thornthwaite (1948) developed dermilk (1934), Hursh and Brater (1944), Hursh (1936) an empirical model for computing monthly evaporation observed subsurface flow as part runoff hydrograph in which is still used. Combining energy balance and mass humid regions. Hoover and Hursh (1943), and Hursh transfer, Penman (1948) developed what is now referred (1944) showed that subsurface storm flow constituted a Singh Geosci. Lett. (2018) 5:15 Page 6 of 18 significant portion of streamflow in humid areas. Rem - Erosion and sediment yield son et al. (1960) and Hewlett (1961a, b) developed con- Cook (1936) identified major factors that impact erosion cepts of source area and partial area that contributed by water. Considering the effect of slope steepness and to streamflow generation and showed that downslope slope length, Zingg (1940) developed an empirical equa- unsaturated flow could contribute to streamside satura - tion for calculating field soil loss. Smith (1941) developed tion and hence generate streamflow. an equation considering additional factors, such as crop- Macropores and preferential flow paths can signifi - ping system and support practices. Browning et al. (1947) cantly contribute to subsurface flow under certain condi - included soil erodibility and management factor in the tions. Germann (1985, 2014) reviewed preferential flow Smith equation. Smith and Whitt (1948) developed an and has given a full account based on the kinematic wave equation as product of average annual soil loss for clay- theory. Macropores are pipe structures in soil matrix and pan soils for a specific rotation, slope length, slope steep - result from physical processes, such as erosion due to ness, and row direction; slope steepness; slope length; soil desiccation cracking and biological activity such as ani- erodibility; and support practice. Musgrave (1947) devel- mal burrows and decaying plant root channels. Tanaka oped an equation considering factors reflecting the effect et al. (1988) found that more than 90% of runoff origi - of rainfall and surface runoff as impacted by slope steep - nated from below the ground mainly through pipe flow. ness and length, and vegetative cover. Using 10,000 plot Leaney et al. (1993) noted that winter stormflow reached years of basic runoff and soil loss data, Wischmeier and the channel primarily through macropores. Newman Smith (1957, 1965, 1978) developed the Universal Soil et al. (1998) inferred that most of the lateral subsurface Loss Equation (USLE) that has undergone several revi- flow occurred in B horizon through macropores. Thus, sions and its new incarnation is Revised USLE (Renard subsurface flow-through macropores and other preferen - et al. 1997). A comprehensive account of soil erosion tial flow paths can be a major contributor to streamflow prediction and prediction is treated in Soil Conservation generation. Society of America (1977). Soil erosion by water was also investigated using Groundwater hydraulic equations. Foster and Meyer (1972) derived an In 1852, Darcy conducted experiments on flow-through equation for sediment transport under steady-state con- sands and developed what is now referred to as Darcy’s dition for rill and inter-rill detachment and/or deposition. law which laid the foundation of quantitative groundwa- Hjelmfelt et al. (1975) considered the kinematic wave for- ter hydrology. Theis (1935) derived the relation between mulation of erosion on a plane. Singh and Regl (1983a, b) drawdown in piezometric head and pump discharge developed the kinematic wave theory for erosion due to from a well. Muskat (1937) published a treatise on flow rainfall. Considering surface flow and rain-drop impact, of homogeneous fluids in porous media. Hubbert (1940) Hairsine and Rose (1992a, b) derived a model for soil ero- described the theory of groundwater motion. Meinzer sion which was based on the equation developed by Rose (1942) edited a book on hydrology. Jacob (1943, 1944) et al. (1983a, b). Both USLE and physically based equa- established the relationship between infiltration and tions of soil erosion have been included in a wide range groundwater. Dynamic changes in streamside ground- of watershed hydrology or erosion models which have water flow were reported by Roessel (1950). Hantush and recently been reviewed by Pandey et al. (2016). Flanagan Jacob (1955) derived equations for unsteady radial flow in and Huang (2017) have provided a review of soil erosion. leaky aquifers. Hantush (1960, 1964) revised the theory of leaky aquifers. Freeze (1975) presented a stochastic con- Sediment transport ceptual analysis of one-dimensional groundwater flow in There is vast literature on sediment transport in reser - nonuniform homogeneous media. The field of groundwa - voirs, rivers and channels that has culminated into a new ter has since expanded dramatically. A large number of field of sedimentation engineering. A number of for - books have been published that detail hydrogeological, mulae have been developed for bed load and suspended scientific, numerical, and engineering aspects of ground - load. The earliest bed load formula was developed by water. Freeze and Cherry (1979) discussed groundwa- DuBoys (1879) assuming uniform grains moving as ter and contamination from a hydrogeology perspective series of layers. Shields (1936) developed a criterion for (Fair and Hatch 1933), Bear (1979) hydraulics of ground- incipient motion of sediment particles. Assuming graded water, Todd (1980) hydrology of groundwater, Domenico sediment, Meyer-Peter and Muller (1948) developed a and Schwartz (1990) physical and chemical hydrogeology formula for bed load sediment transport. With extensive of groundwater, Gelhar (1993) stochastic aspects, and analysis based on fluid mechanics and probability the - Delleur (1999) groundwater engineering. Pham and Tsai ory, Einstein (1942, 1950) developed a bed load function (2017) have reviewed groundwater modeling. for sediment transport in open channels. Brown (1950) Singh Geosci. Lett. (2018) 5:15 Page 7 of 18 modified the Einstein formula. Parker et al. (1982) devel - De Josselin de Jong (1958) developed a random walk oped a bed load equation for coarse-bed material and model for describing longitudinal and transverse disper- gravel-bed rivers. sion in granular materials. (Scheidegger 1961) described Einstein (1950) computed suspended sediment dis- the general theory of dispersion in porous media. Bear charge considering vertical variations in velocity and and Verruijt (1987) presented the theory and applications sediment concentration. Colby (1964) determined bed- of transport in porous media. Palmer (1992) and Fetter material discharge as a function of mean flow velocity, (1999) discussed principles of contaminant hydrogeology depth, mean sediment size, water temperature and con- (Fick 1855). Charbeneau (2000) discussed the hydraulics centration of fine sediment. Using physical laws, Bagnold of groundwater and pollutant transport. Gelhar (1993) (1966) developed an approach for transport of sediment. presented stochastic method in subsurface hydrology. Engelund and Hansen (1967) derived a sediment trans- Agricultural chemicals, fertilizers, weedicides, and port equation using the concept of stream power. Yang pesticides are applied to agricultural fields for increasing (1972) developed a bed-material load equation based on crop productivity. Many chemical compounds generated the rate of energy dissipation of flow. Ackers and White by industries are sometimes dumped on the soil surface. (1973) developed an equation to sediment transport in Sometimes there is a chemical spill on the surface. What- open channel flow as a function of mobility factor. The ever the source or cause, some of the pollutants enter the state of art of sedimentation engineering was provided soil, contaminant it, and percolate down to contaminate by Vanoni (1975). Simons and Senturk (1977) discussed the ground water. Earliest attempts to model solute trans- sediment transport technology. An up-to-date account of port in the unsaturated zone were made by soil scien- sedimentation engineering, including processes, meas- tists. Nielsen and Biggar (1961) discussed a wide range of urements, modeling, and practice, was presented by Gar- problems related to miscible displacement and pollutant cia (2008). Papanicolaou and Abban (2017) have provided transport. Knisel (1980) reported a field-scale model for an up-to-date account of channel erosion and sediment chemicals, runoff, erosion from agricultural management transport, whereas Sarkar (2017) has discussed sedimen- systems, called CREAMS. Leonard et al. (1987) presented tation in floodplains, lakes and reservoirs. a model, called GLEAMS: groundwater loading effects of agricultural systems. Carlsel et al. (1985) developed Pollutant transport a pesticide root zone model (PRZM). Shaffer and Lar - Water quality has always been a major concern but in son (1987) reported a soil–crop simulation model for hydrology it started receiving attention since the 1970s nitrogen, tillage, and crop-residue management, called with the establishment of Environmental Protection NTRM. Smith (1990) described an integrated simulation Agency (EPA). Tremendous work has since been done in model for transport of nonpoint source pollutant at field the hydrology of surface water, vadose zone, and ground- scale, called OPUS. In the 1980s, the U.S. Department of water quality. Both physical and biochemical aspects of Agriculture-Agricultural Research Service reviewed the water quality have been emphasized. Water quality has state of water quality modeling and started to develop been investigated using both systems approach as well a model that would address a wide range of agricul- as science-based approach. In 1925 Streeter and Phelps tural management practices. The resulting model was derived a model for dissolved oxygen in surface waters. Root-Zone Water Quality Model (RZWQM) (RZWQM Taylor (1953, 1954) developed a theory of dispersion of Team 1992) which is a physical, chemical, and biologi- matter in flow in pipes. Elder (1959) determined dis - cal process model and has since undergone a number of persion in turbulent open channels. Fisher (1967, 1968) revisions. This model is more advanced than any of the described mixing in inland and coastal streams. Yot- other models developed before. Zamani and Bombardelli sukura and Sayre (1976) developed a model for transverse (2014) presented analytical solutions for transport of mixing in natural channels. Yotsukura (1977) derived non-reactive species in unsaturated soil. Zamani and equations for solute transport in turbulent natural flow. Ginn (2017) reviewed the state of art of pollutant trans- Thomann (1972) provided a treatise on systems approach port in vadose zone as well as numerical models, includ- to water quality management. Rinaldi et al. (1979) pre- ing SUTRA (Voss and Provost 2002), VS2DT (Healy pared a treatise on river water quality modeling and 1990), HYDRUS (Radcliffe and Simunek 2010), among control. Tchobanoglous and Schroeder (1985) compre- others. hensively discussed water quality characteristics, mod- eling and modification. Thomann and Mueller (1987) Reservoir operation presented principles of surface water quality modeling For reservoir design, operation, and management, water and control. Ji (2008) treated the hydrodynamic modeling surplus, deficit, range, and storage are computed. Two of water quality of rivers, lakes, and estuaries. different tracks, deterministic and stochastic, were Singh Geosci. Lett. (2018) 5:15 Page 8 of 18 pursued for reservoir operation and management. The stochastic methods for water resources systems, includ- deterministic track entailed various optimization tech- ing reservoirs. niques. Indeed these techniques gave birth to the field of water resource systems engineering. One of the earli- Flood frequency analysis est studies in this field was by Mass et al. (1962) under Hazen (1930) presented a treatise on frequency analy- the Harvard water Program. Hall and Dracup (1970) sis of both maximum and minimum flood flows. Foster authored a popular book on water resources systems (1934) derived duration curves. Kendall (1938) derived engineering. With the advent of computers and their a measure of rank correlation. Weibull (1939) presented growing computational power, this field took a giant leap a formula for plotting probability against its quantile. in the 1970s and 1980s. As a result, numerous popular Gumbel (1941) derived a distribution, now called Gumbel books and other publications enriched the literature. A distribution, for frequency analysis of annual maximum sample of books includes those by Haimes (1977), Loucks flows. This distribution is the extreme value type one et al. (1981), and Meta Systems, Inc. (1975). Lund et al. distribution (Boughton 1980). Langbein (1949) analyzed (2017) have provided reservoir operation design. The flood frequencies using partial duration series. Chow optimization techniques employed for analysis and syn- (1951) presented a general formula for frequency analy- thesis of water resources systems allowed to integrate sis based on frequency factor. Jenkinson (1955) derived seemingly disparate areas, such as economics, politics, a general extreme value distribution for frequency analy- decision-making, environmental science, and ecology sis of meteorological data. Gringorten (1963) presented a with hydrology, hydraulics, and water resources engi- formula for plotting positions. neering. u Th s, it was possible to undertake planning of Hershfield (1962) prepared rainfall frequency atlas water resources at the river basin scale. of the United States for durations from 30 min to 24 h The stochastic track assumed that water surplus, defi - and return periods from 1 to 100 years, published as cit, range, and storage needed for reservoir design, opera- U.S. Weather Bureau Technical Report 40, Washington, tion and management varies randomly. Therefore, the D.C. NERC (1975) presented a treatise of flood studies. probability theory was applied to analyze them and com- Houghton (1978) presented the Wakeby distribution pute their probabilities. Three methods have been used for modeling flood flows. Todorovic (1978) developed a for design of reservoirs: empirical, experimental or data methodology for frequency analysis using random num- generation, and analytical. The best example of an empir - ber of random variables. Landwehr et al. (1979, 1980) ical method is the mass curve or Rippl diagram applied developed the probability weighted moments for distri- in England in 1883. The data generation method is also bution parameter estimation. Cunnane (1978, 1989) pro- referred to as Monte Carlo method, synthetic hydrol- vided a review of frequency distributions and presented ogy, or operational hydrology method. Range analysis is a less biased plotting position formula. Hosking (1990) an example of the analytical method. Yevjevich (1972) developed the L-moments method for estimating fre- discussed range analysis. Hurst (1951) investigated long- quency distribution parameters. Dalrymple (1960) devel- term storage capacities of reservoirs which led to what oped a flood index method for regional flood frequency is now known as Hurst coefficient. Thomas and Fiering analysis. Kite (1988) presented different methods of flood (1962) presented a mathematical synthesis of streamflow frequencies and risk analysis. Rao and Hamed (2000) sequences for analysis of river basins. Matalas (1967) provided a comprehensive discussion of flood frequency reported a mathematical assessment of synthetic hydrol- distributions. Stedinger (2017) has presented an up-to- ogy. Mandelbrot and Wallis (1969) performed computer date account of flood frequency distributions and Ouarda experiments with fractional Gaussian noises. Valencia (2017) of regional flood frequency modeling. Vogel and and Schaake (1972) presented disaggregation processes Castellarin (2017) have discussed risk, reliability, and in hydrology. return periods for hydrologic design. The probability theory of reservoir storage or storage theory was developed in the 1950s, although Saverens- Drought analysis kiy (1940) computed probabilities of high and low flows Recent years have witnessed much interest in drought through a probability routing method. Moran (1954) modeling, partly because of the uncertainty about water initiated the storage theory considering serially inde- availability and supply triggered by climate change. pendent reservoir inflows with a fixed probability dis - Many areas in the world are experiencing drought or a tribution. Moran’s theory is based on Markov process. drought-like situation or downright scarcity. Drought Gould (1961) incorporated failures within a year. Lloyd has been defined in different ways. The World Mete - (1963) developed a probabilistic storage theory consider- orological Organization (WMO 1986) defined drought ing serially dependent flows. Kottegoda (1980) discussed as a sustained, extended deficiency in precipitation. The Singh Geosci. Lett. (2018) 5:15 Page 9 of 18 Food and Agriculture Organization (FAO 1983) of the of the 1960s and by the 1970s computers became acces- United Nations defined drought hazard as ‘the percent - sible to universities, government agencies and industry. age of years when crops fail from lack of moisture. Gum- The resulting computing capability made possible the bel (1963) defined drought as the smallest annual value simulation of the entire hydrologic cycle and the birth of daily streamflow, whereas Palmer (1965) described of numerical hydrology. In 1966, Crawford and Lins- drought as a significant deviation from the normal hydro - ley reported the first watershed model, called Stanford logic conditions of an area. Linsley et al. (1959) defined Watershed Model (SWM) that became HSPF (Hydro- drought as a sustained period of time without significant logic Simulation Package-Fortran) in its latter incarna- rainfall. Clearly, the drought definition varies with the tion and BASINS (Better Assessment Science Integrating variable used to define it. Mishra and Singh (2010a) pro - Point and Nonpoint Sources) in its current life. In subse- vided a comprehensive discussion of drought concepts. quent years, a number of models were developed in the Drought modeling encompasses characterization, U.S. Examples of popular ones are HEC-1 (Hydrologic space–time analysis, forecasting, and climate change Engineering Center 1968) which in current form is HEC- impact. The variables associated with drought are pre - HMS (Hydrologic Modeling Simulation), SWMM (Storm cipitation for hydrometeorological drought, streamflow Water Management Model) (Metcalf and Eddy et al. or lake level for hydrologic drought, groundwater level 1971), NWS-RFS (National Weather Service-River Fore- for groundwater drought, and soil moisture for agri- cast System) (Burnash et al. 1973), SSARR (Streamflow cultural drought. The main drought characteristics are Synthesis and Reservoir Regulation) System (Rockwood intensity, duration, severity, and spatial extent. Sev- 1982), and USGS Rainfall–Runoff Model (Dawdy et al. eral indices have been defined, based on combina - 1970) which later became PRMS (Precipitation Runoff tions of precipitation, temperature, soil moisture, and Modeling System) (Leavesley et al. 1983). A large number evapotranspiration, to characterize, assess, and forecast of other hydrology simulation models were developed in droughts. Commonly used indices are: Palmer sever- Australia, Canada, England, Sweden, and other countries. ity drought index (PDSI) (Palmer 1965), Crop Moisture Many of these models are described in Singh (1995), and Index (CMI) (McKee et al. 1993), Soil Moisture Drought Singh and Frevert (2002a, b, 2006). Singh and Woolhiser Index (SMDI) (Hollinger et al. 1993), and Vegetation (2002) appraised the state of art of mathematical mod- Index (VI) (Liu and Kogan, 1996). Also, climatic indices, eling of watershed hydrology. Borah (2011) reviewed and such as El Nino Southern Oscillation (ENSO), South- compared hydrologic procedures of storm-event water- ern Oscillation Index (SOI), Sea Surface Temperature shed models. Donigian et al. (2017) have provided a com- (SST), North Atlantic Oscillation (NAO), Pacific Dec - prehensive discussion of continuous watershed models, adal Oscillation (PDO), Inter-decadal Pacific Oscillation and Gupta and Sorooshian (2017) have discussed the (IPO), and Atlantic Multi-decadal Oscillation (AMO), calibration and evaluation of watershed models. are used for long-lead drought forecasting. Mishra and Singh (2010b) provided a review of drought models that Data observation and tools include regression models, time series models, probabil- Empirical observations form the basis of much of what ity models, artificial neural network models, and hybrid we know about hydrologic systems as well as for their models; and spatio-temporal drought analysis; drought operation and management. For hydrologic modeling, modeling under climate change scenarios. Mishra et al. the types of data needed are hydrometeorologic, physi- (2015) edited a special issue of Journal of Hydrology on ographic, geomorphologic, pedologic, geologic, hydro- drought processes, modeling, and mitigation. Hao et al. metric, land/land cover, and agricultural. Local, state, (2018) reviewed seasonal drought prediction, advances, and federal agencies have been collecting data that are challenges, and future prospects. relevant for their operational and management purposes, but the data so collected have also been and continue to Watershed models be used for research and generating new knowledge. The It is seen that for a period of over a century until the technology for data collection has undergone a revolu- 1960s prior to the computer era, many groundbreaking tionary change over the past three decades in four ways. advances in modeling different components of the hydro - First, data collection tools are much more accurate, such logic cycle were made. Some of these advances were as velocity measurements by acoustic Doppler veloci- based on the laws of mathematical physics and some had metry (ADV). Second, it is now possible to collect data their basis in laboratory and/or field experiments. The that was not possible before, such as direct measure- current state of hydrologic science and engineering owes ment of discharge. Third, it is possible to collect spatial a great deal to the pre-1960 advances. With the advent data rather than point data, such as spatial representa- of computer, the digital revolution started in the decade tion of rainfall field by radar. Fourth, it is now possible Singh Geosci. Lett. (2018) 5:15 Page 10 of 18 to collect data in remote inaccessible areas using satellite data (Singh and Fiorentino 1996). The term geographi - technology. cal information here means the x-, y- and z-coordinates Remote sensing tools, particularly satellites and radar, of land surfaces defined in a coordinate system. Because are becoming more popular these days (Engman and GIS is a data processing tool, tools that provide or record Gurney 1991). Since the launch of Landsat-1 [also known information, such as digital elevation model (DEM), as the Earth Resources Technology Satellite (ERTS)], topographic surveys, land use and land cover maps, can developed by NASA (National Aeronautics and Space be dealt within the GIS environment (Maidment 2002). Administration) and operated by USGS (United States These days, global positioning systems (GPS) and GIS Geological Survey), in 1972, six other satellites have been can be combined to provide more complete information. launched and land surface data have since been collected The use of GIS permits integration of spatial, non-spa - (Shen et al. 2013). The next generation of satellites, called tial, and ancillary data into hydrologic models and thus Landsat Data Continuity Mission (LDMC), was launched significantly strengthens hydrologic modeling capabil - in 2013. Most NASA satellite land measurements can be ity (Mujumdar and Nagesh Kumar, 2012). Griffin et al. found in the NASA Land Measurement Portal (http:// (2017) have comprehensively discussed GIS and their landp ortal .gsfc.nasa.gov) which includes data products applications. in four categories: surface radiation budget, vegeta- tion parameters, land cover/land use changes, and land Tools and methods for analysis hydrosphere. More specifically, one can obtain for hydro - The past half a century has witnessed an unprecedented logic modeling synoptic data of meteorological inputs; development of new tools and techniques for analysis soil and land use parameters; inventories of water bodies, of hydrologic data. Many of these tools were developed lakes, reservoirs, rivers, etc.; snow cover and ice fields; outside of hydrology but they were appropriately tailored and water quality parameters. Other agencies in Japan, for hydrologic applications. Some of these tools include China, and India have also launched spaceborne sensors/ artificial neural networks (Tayfur and Singh 2017), fuzzy missions for studying the terrestrial water cycle compo- logic (Bogardi 2017), genetic algorithms (Kawamura and nents. Examples include Advanced Microwave Scanning Merabtene 2017), relevance vector machines (Tripathi Radiometer (AMSR) and Soil Moisture and Ocean Salin- and Govindaraju 2017), wavelets (Labat 2017), outlier ity (SMOS) for estimating soil moisture; Tropical Rainfall analysis (Panu and Ng 2017), time series analyses (Sveins- Measuring Mission (TRMM) for precipitation; Moderate son and Salas 2017), nonstationarity detection and analy- Resolution Imaging Spectroradiometer (MODIS) for veg- sis (2017), geostatistical methods (Dwivedi et al. 2017), etation; JASON-1 and JASON-2 and TOPEX-POSEIFON generalized frequency distributions (Singh and Zhang for surface water level; and Gravity Recovery and Climate 2017), data assimilation methods (Todini and Biondi Experiment (GRACE) for groundwater and evaporation. 2017), calibration and validation methods (Todini and Lakshmi et al. (2015) presented a treatise on remote sens- Biondi 2017), Bayesian methods (Kuczera et al. 2017), ing of the terrestrial water cycle. Lakshmi (2017) edited a optimization methods (Dozier et al. 2017), nonparamet- book on remote sensing of hydrological extremes. ric methods (Lall and Rajagopalan 2017), uncertainty Weather radar is being employed for spatial mapping of assessment and decision-making (Todini 2017), risk and rainfall field and daily weather forecasting. Both ground- reliability analysis (Tung and Mays 2017), scaling and based and spaceborne radars are used. With the use of fractals (Veneziano and Lepore 2017), chaos theory (Siva- bias correction techniques, radar rainfall data are usually kumar 2017), copula theory (Genest and Chebana 2017), scaled to match data being observed at rainfall gauging entropy theory (Singh 2013, 2014, 2015, 2016, 2017a, c), stations. Even though radar rainfall data in many cases data mechanistic modeling (Young 2017), decomposition are available on web, their use with quality control/assur- methods (Serrano 2017), and network theory (Sivakumar ance and bias correction is recommended. Pathak et al. et al. 2017). These techniques have greatly contributed to (2017) edited a special issue of Journal of Hydrologic not only the increased understanding of hydrologic sys- Engineering on radar rainfall and operational hydrol- tems but also hydrologic practice. ogy that contains papers dealing with radar rainfall data estimation, improvement, and validation; application Emerging areas of radar rainfall data; and use of radar rainfall for flood Many new areas have merged during the past couple of forecasting. decades and others will emerge in the decades ahead. Hydrology of global warming and climate change is an Geographical information systems area that has been receiving a lot of attention in public Geographical information systems (GIS) are a technology fora, primarily because of increased frequency of hydro- for stacking, analyzing, and retrieving large amounts of meteorologic extremes and significant variability in the Singh Geosci. Lett. (2018) 5:15 Page 11 of 18 space–time distribution of precipitation (McCuen 2017). and tailor it to produce leaders of tomorrow who will be Ecosystem hydrology is another area that has recently well equipped to address the societal needs of tomorrow. emerged. Hydrologic impacts of hydraulic fracturing Likewise, research funding agencies will have to rethink are in much public debate these days. Transport of bio- and reprioritize their direction of funding in concert with chemical and microorganisms is receiving plenty of trac- these grand challenges and pressing societal needs. tion. Hydrology of hurricanes and typhoons is a newly Social or rural hydrology, extraterrestrial water, water emerging area. Atmospheric rivers are receiving much and food and energy security are newly emerging areas. attention. Hydrology of long-distance water transfer is For management of hydrologic systems, political, eco- receiving global attention these days. Hydrology has a nomic, legal, social, cultural, and management aspects value to society and a new area, called social hydrology, will need to be integrated. It is vital that both hydrologic has lately emerged and is getting traction in scientific science and engineering applications are equally empha- discourses. sized. Hydrologic science must not be allowed to be over- taken by data cranking methods borrowed from outside. Integration of concepts and processes At the same time, data analysis tools must be seamlessly Because of computing prowess and sophisticated instru- integrated with hydrologic science. mentation available these days, integration in and across hydrology is occurring rapidly. Hydrology and climatol- Conclusions ogy are being integrated and hydroclimatology is emerg- The following conclusions are drawn from this study: ing with renewed emphasis. Ecology and hydrology have combined to give birth to ecohydrology. Likewise, coastal 1. Hydrologic modeling has come a long way from its science and hydrology are being integrated leading to modest beginning in the 1850s. Advances in mod- coastal hydrology. The field of hydrology is broadening eling have occurred at an increasing pace, primarily and the areas, such as social science, culture and religion, driven by easy access to almost limitless computing politics, economics, and health sciences are being inter- capability, sophisticated instrumentation, and remote faced with hydrologic sciences. Greater integration of sensing and GIS capabilities. concepts from intelligent systems, software engineering, 2. Integration of hydrology with allied areas is occur- information engineering, and humanities is envisioned in ring increasingly and will so continue. the years ahead. 3. The role of hydrology is coming into sharper focus, because of global warming and climate change on Future outlook one hand and water, food and energy security on the With advances in data capturing and analysis capabilities other hand. and information technologies, it seems that the future of 4. Information technology is being assimilated in hydrology will be even brighter. It can be expected that hydrology without much resistance. new tools will be at the disposal of hydrology. For exam- 5. Hydrology is receptive in adopting techniques being ple, drones will become commonplace for acquiring spa- developed in mathematics, statistics, and sciences. tial data. Hydrologic models will become so user-friendly Authors’ contributions that little hydrologic knowledge will be needed to oper- VPS conceptualized the framework and crafted the manuscript. The author ate them, just like one does not need to be an automobile read and approved the final manuscript. engineer to drive a car or an electrical engineer to oper- ate an electrical system. Each model, however, simple or Acknowledgements complicated, will be associated with a statement of uncer- The author greatly appreciates the invitation extended by Professor B. Siva- kumar, University of New South Wales, Australia, to prepare this paper and is tainty. New frontiers of hydrology will unfold with the grateful for the support extended from time to time. use of cell phones and newly emerging information tech- nologies. Hydrologic forecasting capability will multiply. Competing interests The author declares that he has no competing interests. There will be greater interaction between the user and the model and the modeler. This has already started hap - Availability of data and materials pening through what is now regarded as social hydrology. Not applicable. Hydrology will play an increasing role in meeting grand Consent for publication challenges of this century, such as water security, food The author consents for publication. security, energy security, environmental security, health Ethics approval and consent to participate security, food–water–energy nexus, and sustainable Not applicable. development. These grand challenges will also compel educators to revisit the delivery of hydrologic education Singh Geosci. Lett. (2018) 5:15 Page 12 of 18 Funding Budyko MI (1955) On the determination of evaporation from the land surface. No funding source is available. Meteorol Gidrol 1:52–58 (Russian) Budyko MI (1974) Climate and life. Int Geophys Ser 18:508 Bultot FG, Dupriez DL, Bodeaux A (1972) Interception of rain by forest vegeta- Publisher’s Note tion: estimation of daily interception by using mathematical models. J Springer Nature remains neutral with regard to jurisdictional claims in pub- Hydrol 17(3):193–223 lished maps and institutional affiliations. Burnash RJC, Ferral RL, McGuire RA (1973) A generalized streamflow simula- tion system-conceptual modeling for digital computers. 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California water recharge in sandy sediments at Seabrook, New Jersey. Soil Sci Institute of technology, Pasadena, California, p 21 89:145–156 Shreve RL (1966) Statistical law of stream numbers. J Geol 74:17–37 Remson I, Hornberger GM, Molz FJ (1971) Numerical methods in Subsurface Simons DB, Senturk F (1977) Sediment transport technology. Water Resources Hydrology. John Wiley, New York, p 389 Publications, Littleton, Colorado, p 897 Renard KG, Foster GR, Weesies GA, McCool DK, Yoder DC (1997) Predicting soil Singh VP (1988) Hydrologic systems: rainfall-runoff modeling, vol 1. Prentice erosion by water: a guide to conservation planning with the revised hall, Englewood Cliffs universal soil loss equation (RUSLE). U.S. Department of Agriculture, Singh VP (1989) Hydrologic systems: watershed modeling, vol 2. Prentice hall, Agriculture Handbook No. 703, Washington, p 404 Englewood Cliffs Richards LA (1931) Capillary conduction of liquids in porous mediums. Physics Singh VP (1992) Elementary hydrology. Prentice Hall, Englewood cliffs 1:318–333 Singh VP (1994) Accuracy of kinematic wave and diffusion wave approxima- Richards LA (1965) Physical conditions of water in soils. In: Black CA (ed) tions for space independent flows. Hydrol Process 8(1):45–62 Methods of soil analysis, monograph 9. American Society of Agronomy, Singh VP (ed) (1995) Computer models of watershed hydrology. Water Madison, pp 128–151 Resources Publications, Littleton Richardson B (1931) Evaporation as a function of insolation. Trans Am Soc Civ Singh VP (1996) Kinematic wave modeling in water resources: surface water Eng 95:996–1011 hydrology. John Wiley, New York Rinaldi S, Soncini-Sessa R, Stehfest H, Tamura H (1979) Modeling and control of Singh VP (1997) Kinematic wave modeling in water resources: environmental river quality. McGraw-Hill Book Publishing Company, New York hydrology. John Wiley, New York Rockwood DM (1982) Theory and practice of the SSARR model as related to Singh VP (2003) On the theories of hydraulic geometries. Int J Sedim Res analyzing and forecasting the response of hydrologic systems. In: Singh 18(3):196–218 VP (ed) Applied modeling in catchment hydrology. Water Resources Singh VP (2013) Entropy theory and its application in environmental and water Publications, Littleton, pp 87–106 engineering. John Wiley, New York, p 642 Singh Geosci. Lett. (2018) 5:15 Page 17 of 18 Singh VP (2014) Entropy theory in hydraulic engineering. ASCE Press, Reston Smith DD, Whitt DM (1948) Evaluating soil losses from filed areas. Agric Eng Singh VP (2015) Entropy theory in hydrologic science and engineering. 29:394–398 McGraw-Hill Education, New York Smith RE, Woolhiser DA (1971) Overland flow on an infiltrating surface. Water Singh VP (2016) Introduction to tsallis entropy theory in water engineering. Resour Res 7(4):899–913 CRC Press/Taylor & Francis Group, Boca Rtaon, Florida, p 434 Smith RE, Smetten KRJ, Broadridge P, Woolhiser DA (2002) Infiltration theory Singh VP (ed) (2017a) Handbook of applied hydrology. McGraw-Hill Education, for hydrologic applications. Water resources monograph 15. American New York Geophysical Union, Washington, D. C, p 212 Singh VP (2017b) Entropy theory. Chapter 31. In: Singh VP (ed) Handbook of Soil Conservation Service (1956) National engineering handbook, supple- applied hydrology. McGraw-Hill Education, New York, pp 31-1–31-8 ment A, section 4, hydrology, chapter 10. Department of Agriculture, Singh VP (2017c) Kinematic wave theory of overland flow. Water Resour Man- Washington, D. C age. https ://doi.org/10.1007/s1126 9-017-1654-1 Soil Conservation Society of America (1977) Soil erosion: prediction and con- Singh VP, Fiorentino M (eds) (1996) Geographical information systems in trol. Soil Conservation Society of America, Ankeny, p 393 hydrology. Kluwer Academic Press, Dordrecht Sorooshian S, Hsu K-L, Coppola E, Tomasseti B, Verdecchia M, Visconti G (eds) Singh VP, Frevert DK (eds) (2002a) Mathematical models of large watershed (2008) Hydrological modeling and the water cycle: coupling the atmos- hydrology. Water Resources Publications, Littleton pheric and hydrologic models. Springer, Dordrecht, p 291 Singh VP, Frevert DK (eds) (2002b) Mathematical model of small watershed Stebbings J (1963) The shape of self-formed model alluvial channels. In: Pro- hydrology and applications. Water Resources Publications, Littleton ceedings of the Institute Civil Engineers, London, 25: 485–510 Singh VP, Frevert DK (eds) (2006) Watershed models. CRC Press-Taylor and Stedinger JR (2017) Flood frequency analysis. Chapter 76. In: Singh VP (ed) Francis, Boca Raton Handbook of applied hydrology. McGraw-Hill, New York, pp 76-1–76-8 Singh VP, Regl RR (1983a) Analytical solutions of kinematic equations for Stoker JJ (1953) Numerical solution of flood prediction and river regulation erosion on a plane: I. Rainfall of indefinite duration. Adv Water Resour problems: 1. Derivation of basic theory and formulation of numerical 6:1–10 methods of attack. Report IMM-200, Institute of Mathematical Sciences, Singh VP, Regl RR (1993b) Analytical solutions of kinematic equations for ero- New York University, New York sion on a plane: II. Rainfall of finite duration. Adv Water Resour 6:88–95 Strahler AN (1952) Dynamic basis of geomorphology. Geol Soc Am Bull Singh VP, Woolhiser DA (2002) Mathematical modeling of watershed hydrol- 63:923–938 ogy. J Hydrol Eng 7(4):270–294 Strahler AN (1957) Quantitative analysis of watershed geomorphology. Trans Singh VP, Yu FX (1990) Derivation of an infiltration equation using systems Am Geophys Union 38:913–920 approach. J Irrig Drain Eng 116(6):837–858 Streeter HW, Phelps EB (1925) A study of the pollution and natural purification Singh VP, Zhang L (2008a) At-a-station hydraulic geometry relations, 1: theoret- of the Ohio River. U.S. Public Health Bulletin, 146, February ical development. Hydrol Process 22:189–215 Sveinsson OGB, Salas JD (2017) Time series analysis and models Chapter 18. In: Singh VP, Zhang L (2008b) At-a-station hydraulic geometry relations, 2: calibra- Singh VP (ed) Handbook of applied hydrology. McGraw-Hill Education, tion and testing. Hydrol Process 22:216–228 New York, pp 18-1–18-11 Singh VP, Zhang L (2017) Frequency distributions Chapter 21. In: Singh VP (ed) Tanaka T, Yasuhara M, Sakai H, Marui A (1988) Then Hachioji experimental basin Handbook of applied hydrology. McGraw-Hill Education, New York, pp study-storm runoff processes and the mechanism of its generation. J 21-1–21-11 Hydrol 102:139–164 Singh VP, Zhang L (2018) Copula-entropy theory for multivariate stochas- Tayfur G (2012) Soft Computing in water resources engineering: artificial tic modeling in water engineering. Geosci Lett 5(6):17. https ://doi. neural network, fuzzy logic and genetic algorithms. WIT Press, South- org/10.1186/s4056 2-018-0105-z ampton, p 267 Singh VP, Bengtsson L, Westerstrom G (1997a) Kinematic wave modeling of Tayfur G, Singh VP (2017) Artificial neural networks Chapter 11. In: Singh VP vertical movement of snowmelt water through a snowpack. Hydrol (ed) Handbook of applied hydrology. McGraw-Hill Education, New York, Process 11:149–167 pp 11-1–11-6 Singh VP, Bengtsson L, Westerstrom G (1997b) Kinematic wave modeling of Taylor GI (1953) Dispersion of soluble matter in solvent flow flowing slowly saturated basal flow in a snowpack. Hydrol Process 11:177–187 through a tube. Proc R Soc London Series A 219:186–203 Singh VP, Yang CT, Deng ZQ (2003a) Downstream hydraulic geometry rela- Taylor GI (1954) The dispersion of matter in turbulent flow through a pipe. Proc R tions: 1. Theoretical development. Water Resour Res 39(12):1337. https Soc London Series A 223:446–468 ://doi.org/10.1029/2003W R0024 84 Tchobanoglous G, Schroeder ED (1985) Water quality: characteristics, mod- Singh VP, Yang CT, Deng ZQ (2003b) Downstream hydraulic geometry rela- eling, and modification. Addison-Wesley Publishing Company, Reading, tions: 2. Calibration and testing. Water Resour Res 39(12):1338. https :// p 768 doi.org/10.1029/2003W R0024 98 Theis CV (1935) The relation between the lowering of the piezometric surface Singh VP, Jain SK, Tyagi A (2007) Risk and reliability analysis. ASCE Press, Reston, and the rate and duration of discharge of a well using ground-water p 783 storage. Trans Am Geophys Union 16:519–524 Singh VP, Singh P, Haritashya UK (2011) Encyclopedia of snow, ice and glaciers. Thomann RV (1972) Systems analysis and water quality management. Dordrecht, Springer McGraw-Hill Book Company, New York, p 286 Sivakumar B (2017) Nonlinear dynamics and chaos. Chapter 29. In: Singh VP Thomann RV, Mueller JA (1987) Principles of surface water quality modeling (ed) Handbook of applied hydrology. McGraw-Hill Education, New York, and control. Harper & Row, Publishers, New York, p 644 pp 29-1–29-11 Thomas HA Jr, Fiering MB (1962) Mathematical synthesis of streamflow Sivakumar B, Berndtsson R (2010) Advances in data-based approaches for sequences for analysis of river basins by simulation. In: Mass A et al hydrologic modeling and forecasting. World Scientific, Singapore, p 519 (eds) The design of water resources systems. Harvard University Press, Sivakumar B, Woldemeskel FM, Singh VP (2017) Network theory Chapter 35. In: Cambridge, pp 459–493 Singh VP (ed) Handbook of applied hydrology. McGraw-Hill Education, Thornthwaite CW (1948) An approach toward a rational classification of New York, pp 35-1–35-10 climate. Geogr Rev 38:55–94 Smart JS, Surkan AJ (1967) The relation between mainstream length and area Todd DK (1980) Ground water hydrology. John Wiley, New York in drainage basins. Water Resour Res 3:963–974 Todini E (2017) Predictive uncertainty assessment and decision making Chap- Smith DD (1941) Interpretation of soil conservation data for field use. Agric ter 25. In: Singh VP (ed) Handbook of applied hydrology. McGraw-Hill Eng 22:173–175 Education, New York, pp 25-1–25-16 Smith TR (1974) A derivation of the hydraulic geometry of steady-state chan- Todini E, Biondi D (2017) Calibration, parameter estimation, uncertainty, data nels from conservation principles and sediment transport laws. J Geol assimilation, sensitivity analysis, and validation Chapter 22. In: Singh VP 82:98–104 (ed) Handbook of applied hydrology. McGraw-Hill Education, New York, Smith RE (1990) OPUS: an integrated simulation model for transport of non- pp 22-1–22-19 point source pollutants at field scale, vol 1. Documentation. USDA-ARS Todorovic P (1978) Stochastic models of floods. Water Resour Res Report 98, U.S. Department of Agriculture, Fort Collins, Colorado, p 120 14(2):345–356 Singh Geosci. Lett. (2018) 5:15 Page 18 of 18 Tripathi S, Govindaraju RS (2017) Relevance vector machines. Chapter 14. In: Wischmeier WH, Smith DD (1957) Factors affecting sheet and rill erosion. Trans Singh VP (ed) Handbook of applied hydrology. McGraw-Hill Education, Am Geophys Union 38:889–896 New York, pp 14-1–14-7 Wischmeier WH, Smith DD (1965) Predicting rainfall-erosion losses from Tung Y, Mays LW (2017) Risk-reliability analysis Chapter 27. In: Singh VP (ed) cropland east of the Rocky Mountains: guide for selection of practices Handbook of applied hydrology. McGraw-Hill Education, New York, pp for soil and water conservation. Department of Agriculture (USDA) 27-1–27-10 Agriculture Handbook, Washington, p 282 Tung Y-K, Yen BC (2005) Hydrosystems engineering uncertainty analysis. Wischmeier WH, Smith DD (1978) Predicting rainfall erosion losses: a guide to McGraw-Hill, New York, p 273 conservation planning. Department of Agriculture (USDA) Agriculture Turner LB (1967) Abstraction of depression storage from storms on small Handbook, Washington, p 537 impervious areas. Unpublished M.S. Thesis, University of Maine, Orono, WMO ( World Meteorological Organization) (1986) Report on drought and Maine countries affected by drought during 1974–1985. WMO, Geneva, p 118 Ullah W, Dickinson WT (1979a) Quantitative description of depression storage Wolman MG (1955) The natural channel of Brandywine Creek, Pennsylvania. model using a digital surface model: I. Determination of depression U.S. Geological Survey Professional Paper 271, Washington, D. C storage. J Hydrol 42:63–75 Woolhiser DA, Liggett JA (1967) Unsteady one-dimensional flow over a plane: Ullah W, Dickinson WT (1979b) Quantitative description of depression stor- the rising hydrograph. Water Resour Res 3(3):753–771 age model using a digital surface model: II. Characteristics of surface Yalin MS, Da Silva AMF (1997) On the computation of equilibrium channels in depressions. J Hydrol 42:77–99 cohesionless alluvium. J Hydrosci Hydraul Eng 15(2):1–13 U.S. Army Corps of Engineers (1936) Method of flow routing. Report on survey Yalin MS, Da Siva AMF (1999) Regime channels in cohesionless alluvium. J for flood control, Connecticut River Valley, vol 1, section 1, Appendix, Hydraul Res 37(6):725–742 Providence, Rhode Island Yang CT (1971) Potential energy and stream morphology. Water Resour Res U.S. Army Corps of Engineers (1956) Snow hydrology. Summary report of the 7(2):311–322 snow investigations. North Pacific Division, Portland, Oregon Yang CT (1972) Unit stream power and sediment transport. J Hydraul Div ASCE Valencia RD, Schaake JC Jr (1972) Disaggregation processes in stochastic 18(HY10):1805–1826 hydrology. Water Resour Res 9(3):291–295 Yang CT, Song CCS (1986) Theory of minimum energy and energy dissipa- Vanoni V (ed) (1975) Sedimentation engineering. ASCE manuals and reports tion rate, Chapter 11. In: Cheremisinoff NP (ed) Encyclopedia of fluid on engineering practice no. 54. American Society of Civil Engineers, mechanics, Gulf Publish. Company, Houston New York, (now Reston, Virginia), p 745 Yevjevich V (1972) Stochastic processes in hydrology. Water Resources Publica- St. Venant de B (1871) Theory of unsteady water flow, with application to river tions, Highlands Ranch, p 276 floods and to propagation of tides in river channels. French Academy of Yotsukura N (1977) Derivation of solute-transport equation for a turbulent Science, vol 73, pp 148–154, 237–240 natural-channel flow. J Res US Geol Survey 5(3):277–284 Veneziano D, Lepore C (2017) Scaling and fractals Chapter 28. In: Singh VP (ed) Yotsukura N, Sayre WN (1976) Transverse mixing in natural channels. Water Handbook of applied hydrology. McGraw-Hill Education, New York, pp Resour Res 12(4):695–704 28-1–28-6 Young P (2017) Data-based mechanistic modeling Chapter 33. In: Singh VP Vogel RM, Castellarin A (2017) Risk, reliability, and return periods and hydro- (ed) Handbook of applied hydrology. McGraw-Hill Education, New York, logic design. Chapter 78. In: Singh VP (ed) Handbook of applied hydrol- pp 33-1–33-12 ogy. McGraw-Hill, New York, pp 78-1–78-10 Zamani K, Bombardelli FA (2014) Analytical solutions of nonlinear and variable- Voss CI, Provost AM (2002) SUTRA, a model for saturated-unsaturated variable parameter transport equations for verification of numerical solvers. density groundwater flow with energy or solute transport. U.S. Geologi- Environ Fluid Mech 14:71–742 cal Survey Water Resources Investigations Report 2002–4231. Reston, Zamani K, Ginn TR (2017) Pollutant transport in vadose zone Chapter 68. In: Virginia, p 291 Singh VP (ed) Handbook of applied hydrology. McGraw-Hill Education, Wagner T, Wheater HS, Gupta HV (2004) Rainfall-runoff modelling in gauged New York, pp 68-1–68-8 and ungauged catchments. Imperial College Press, London, p 306 Zeeman EC (1978) Catastrophe theory. Addison Wesley, Boston, p 674 Weibull W (1939) The phenomenon of rupture in solids. Ingeniors Vertenskaps Zingg AW (1940) Degree and length of land slope as it affects soil loss in Akademien Handlinga 153:17 runoff. Agric Eng 21:59–64 White WR, Bettess R, Paris E (1982) Analytical approach to river regime. J Hydraul Div Proc ASCE 108(HY10):1179–1193
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