Methods to estimate evaporation from natural land surfaces can generally be subdivided into three broad categories. Mass transfer methods make use of measurements of wind velocity, temperature, and humidity; energy-budget methods require the same measurements in addition to measurements of radiation and heat conduction into the ground; water-budget methods rely on inflow, outflow, and storage change measurements in control volumes in the lower atmosphere, in the near surface soil, or in entire watersheds.

Most near-surface geologic formations which contain water are unconsolidated porous rocks, broadly referred to as soils close to the surface, and as aquifers at greater depths. The relationship between the degree of water saturation of such a porous material and pressure of the water is referred to as the soil-water characteristic. The specific flow rate of water can usually be assumed to be proportional to the hydraulic gradient, according to Darcy’s law. The proportionality constant, called the hydraulic conductivity, generally exhibits anisotropy and scale dependency, and is a strong function of the degree of water saturation. Although some insight can be gained from theoretical estimation models, it is best determined by experiment. For certain problems it can be convenient to transform Darcy’s law into a diffusion equation, by making the flow rate proportional to the water content gradient. For rigid porous media, combination of the continuity equation with Darcy’s law yields the Richardson-Richards equation; under steady saturated conditions this becomes the Laplace equation. For elastic saturated porous media this combination leads to the Terzaghi and Jacob equations.

Given the easy embodiment of water vapor in air and its short residence times, the lower atmosphere is one of the critical pathways in the global hydrologic cycle; it transports water and energy around the globe without regard to continental boundaries and thus links the continents, the upper atmosphere, and the oceans. The transport and distribution of water vapor in the lower atmosphere, where it is most abundantly present, are among the main factors controlling precipitation and evaporation from the surface; these processes, in turn, determine soil and groundwater storage, and the different runoff phenomena. For purposes of practical analysis, the lower atmosphere can be treated as a turbulent boundary layer, allowing the application of similarity techniques to describe transport not only of water vapor, but also of momentum and sensible heat. The magnitudes of these transport phenomena and their interactions in the lower atmosphere are constrained by the surface energy budget as a critical boundary condition.

Early prehistoric accounts of water cycling in nature refer only to, or hint at, the atmospheric phase of the water cycle. Wherever evaporation is alluded to, it is mostly assumed to take place from rivers and the sea. Speculations on the origin of these streams or on whether or how their water returns to where the streams originated, came later in Greek antiquity. This era produced essentially four competing theories on this, namely the seawater filtration theory, the underground condensation theory, the concept of pre-existing underground primal water, likely based on mythology and less accepted by the philosophers, and the rainfall percolation theory. Although the latter contains the essence of our present understanding, it took nearly another 23 centuries before it became the only remaining one to be fully accepted. In recorded history it can be followed as a thread running through the works of the pre-Socratics, the post-Aristotelian Peripatetics, Vitruvius in ancient Rome, Buridan and other medieval Schoolmen, Bartas, Palissy, and Gassendi in the Renaissance, Mariotte, Ray, and Van Musschenbroek at the dawn of modern science, and finally Dalton in the early nineteenth century.

For some purposes, the physical processes relating current runoff to precipitation can best be assumed to take place at the scale of the catchment, without consideration of the detailed subscale processes or for the intricate flow paths inside the watershed. The most common implementation of this idea has been the unit hydrograph (UH), which is based on the assumptions of linearity and stationarity. A UH is characterized by the duration of its precipitation input; this allows the definition of the instantaneous UH, that is the response of a catchment to a delta function precipitation input, or its Green’s function. The UH of a catchment can be identified from available data using the method of least squares. To facilitate the concise parameterization of UH functions for identification and prediction purposes, various conceptualizations have been proposed consisting of different combinations of linear translation elements and linear storage elements. Attempts have been made to extend the UH concept by allowing for nonstationarity and nonlinearities in the response. Long-term streamflow response to mean annual precipitation has also been the subject of many studies.

In hydrology it is often necessary to assign a probability to future occurrence of an event of a given magnitude, on the basis of an available record of measurements. While general probability theory provides the basis, over the years some concepts have been developed especially as tools in hydrology. A rough estimate of an event’s nonexceedance probability can be derived from its plotting position in the record; a display of the data on probability graph paper is a useful additional tool. It is often useful to fit a mathematical probability distribution function to the available data, because this provides a succinct description of the data and it allows the formulation of objective confidence criteria. Most probability functions have found application in hydrology. The normal distribution is generally appropriate for long-term averages. The log-gamma (or log-Pearson Type III) distribution is now the preferred function for annual maximal river flows. Several extreme value distributions can describe the smallest and largest extremes of different hydrologic phenomena. Methods have been developed to extend regular data records by inclusion of historical events and by regionalization.

In the analysis of most free-surface flows in hydrology it can be assumed that the pressure distribution is hydrostatic normally to the bottom; this in turn allows the adoption of a uniform velocity profile. These two simplifications form the basis of shallow-water theory. In the resulting continuity and momentum equations, also referred to as the Saint Venant equations, the effects of viscosity and turbulence are parameterized in terms of a friction slope. These equations are not easy to solve in general, but important features of free-surface flow can be brought out by solutions of their linearized, diffusion, quasi-steady-uniform flow (or kinematic wave), and lumped kinematic approximations.

As the hydrologic cycle is driven by it, precipitation must be considered its main component: without precipitation, there is also not much of a hydrologic cycle. Precipitation naturally follows supersaturation of the air, usually as a result of cooling. One of the most effective ways of cooling occurs through lifting of the air mass, often involving a cyclonic motion. Most precipitation weather systems can be classified according to their type of cyclonic motion. For many hydrologic applications it is necessary to consider not only the spatial but also the temporal distribution of the precipitation, and many procedures are available for this purpose. The part of precipitation that moistens the surface elements, and is temporarily stored on them, is referred to as interception; often also called interception loss, it can amount to as much as 30 to 40% of the precipitation in dense forests. Detailed energy-budget considerations show that snow melt is mainly driven by the air temperature above freezing. Most past records of precipitation suffer from substantial systematic error. The main factor is wind. Different measurement techniques have been proposed to solve this problem.

Streamflow routing describes the motion of a flood wave in a well-defined open channel. Two extreme types of large waves can be discerned, depending on the main factors controlling the momentum budget in the shallow-water equations. An abrupt wave, a surge or moving hydraulic jump occurs when the inertia and hydrostatic pressure gradient terms are predominant and the friction and gravity terms can be neglected; such types of waves have caused disastrous floods. The monoclinal rising wave occurs in the opposite situation, when friction and slope terms predominate compared to the dynamic terms. Most flood waves are intermediate and their analysis requires, beside the continuity equation, inclusion of the complete momentum equation. Yet, in practice excellent results have been obtained with lumped kinematic methods, among which the well-known Muskingum method, which uses only the continuity equation without explicit momentum conservation considerations. A kinematic approach normally leaves the shape of the flood wave unchanged. However, numerical diffusion due to the discretization of the continuity equation in finite increments allows the description of the changing shape of the wave.

Base flow is the rate of flow that a given river basin can sustain in the absence of precipitation and artificial storage works. Such flows are important in connection with water supply and water quality in rivers during drought periods, and general basin and agricultural drainage. But even storm runoff is largely supplied into the streams by subsurface transport. Thus, subsurface drainage from the aquifers along the banks of the streams is one of the key elements in catchment hydrology, not only under drought conditions but also in response to precipitation. Herein the subsurface outflow is first considered locally at the point where it enters the stream, by analysis of the groundwater flow process in the riparian unconfined aquifer. Thus, the different available formulations are reviewed; these comprise general unconfined flow, free-surface flow, and hydraulic groundwater theory and its linear approximation, including flow in sloping aquifers. In the last section, the base flow is parameterized at the basin scale, by integration of the local outflows along all the streams in the basin. The base flow trend of a basin provides an indication of its groundwater storage changes.

Streamflow is normally characterized by a hydrograph, which is the flow rate as a function of time and is the integrated result of all upstream flow processes. In the earlier chapters some of the more important transport mechanisms have been considered, and most of them are fairly well understood individually. However, there is still no unifying theory available that provides a coherent explanation for the integration of these different local mechanisms into the streamflow generation process. The main reason for this uncertainty is the large variation in drainage basins; but even for any given basin, it is difficult to identify the different mechanisms by decomposing of the runoff integral into its constituent parts, that is by its inversion to obtain the integrands. In brief, the main mechanisms are overland flow, as infiltration excess or as saturation excess, and subsurface storm flow, as regular porous media flow, with or without macropores and other preferential flow paths, shallow permeable layers, or wave-like water-table motion. The stucture of the models and their parameters used in formulating these mechanisms depend on the adopted computational scale.

Infiltration is the movement of water into an unsaturated soil profile. When the precipitation is intense enough to saturate the soil surface, part of it remains ponded or runs off, and part of it infiltrates at the maximal rate; this is the infiltration capacity. Initially the infiltration capacity is mainly driven by the pressure gradient due to capillarity and the effect of gravity is relatively small; thus for small times the problem can be treated as one of horizontal flow or simple sorption. For larger times the effect of gravity gradually takes over, and the infiltration rate approaches the saturated hydraulic conductivity. When the precipitation intensity is low, initially all the water seeps into the pores; this is rainfall infiltration. As precipitation continues, the increasing surface soil-water content is unknown, but the flow at the surface is the known precipitation rate. In time, the soil at the surface becomes saturated, after which water ponds the surface. Important parts of the solution are the time to ponding and the ensuing infiltration rate. Many problems of evaporation from soils can be analyzed on the basis of the isothermal flow equation, viz. Darcy’s law.

Hydrology deals mainly with the cycling of water in the natural environment, as related specifically to the continental water processes, at all scales. Globally, precipitation on the land surfaces averages roughly 0.8 m per year, of which around 60% is evaporated back into the atmosphere and 40% runs off into the oceans. Beside statistical analysis of direct measurements, several methodologies are available to describe the various components of the hydrologic cycle. These can be distinguished from one another primarily by the spatial and temporal scales at which the particular hydrologic phenomena are parameterized.