7 A history of hydrology

Hydrology, the science of water on and under the land surface and its exchange with the atmosphere, grew from the same roots as hydraulics but followed its own path. Its central idea, the hydrological cycle, is simple to state: water evaporates, falls as precipitation, flows over and through the land, and returns to the sea. Yet for more than two thousand years many learned people believed that rivers were fed mainly by seawater rising through the Earth, and it took careful measurement in the late 17th century to show that rainfall is enough. This chapter traces how hydrology became a quantitative science: from rain gauges and river stages to the water balance, from empirical formulas for floods to physical models of evaporation and runoff, and from local engineering to a global science of a changing planet.

Measuring water in the ancient world¶

The first hydrological records were kept for practical reasons: to predict harvests, collect taxes and manage floods.

Nilometers. In Egypt, the annual flood of the Nile was measured with graduated stone steps or wells connected to the river, some dating back to the Old Kingdom. The height of the flood determined the harvest and the tax to be collected. The nilometer on Roda Island in Cairo, rebuilt in 861 CE, has records of annual flood levels that span many centuries and have been used by modern scientists to study long-term variability.
Rain gauges in India. The Arthashastra, a treatise on statecraft attributed to Kautilya (around the 4th century BCE, though compiled over several centuries), describes the measurement of rainfall in a bowl of standard size and gives typical amounts for different regions as a basis for agriculture and taxation.
The water balance in China. Chinese administrators recorded rainfall and floods for agricultural planning; a mathematical treatise of 1247 by Qin Jiushao includes problems on calculating rainfall from gauges of different shapes.

Korea’s rain gauge and river gauge¶

One of the most remarkable early measurement networks was created in Korea. In 1441, under King Sejong of the Joseon dynasty, the cheugugi (측우기), a standardized bronze rain gauge, was introduced, and in 1442 rules were issued for installing gauges in the capital and in provincial offices across the country, with rainfall to be measured in standard units and reported to the court. At the same time, a graduated stone pillar, the supyo (수표), was set up in the Cheonggyecheon stream in Seoul to measure the water level, together with a gauge in the Han River.

The network declined after the Japanese invasions of the 1590s and was revived in 1770 under King Yeongjo. From 1777 to 1907, daily rainfall in Seoul was recorded almost continuously, one of the longest instrumental precipitation records in the world. Climate scientists have used it to study the long-term variability of the East Asian monsoon.

Where do springs come from?¶

Ancient philosophers recognized parts of the hydrological cycle. Aristotle, in his Meteorologica, described the evaporation of water by the Sun and its condensation into rain, but he and most of his successors doubted that rain could supply all rivers and springs. They imagined instead that air turned into water inside cold caverns, or that seawater flowed through underground channels into the mountains, losing its salt along the way. Plato wrote of a great underground reservoir, Tartarus, from which all rivers flow. The Roman architect Vitruvius came closer to the truth, arguing that rain and snow melting in the mountains seep into the ground and emerge as springs at lower levels.

The question remained open until the Renaissance. Leonardo da Vinci thought about the circulation of water through the Earth, comparing it to the circulation of blood. The French potter and naturalist Bernard Palissy (around 1510–1590) argued in 1580 that springs and rivers are fed entirely by rainwater infiltrating the ground, but he gave no measurements.

The birth of quantitative hydrology: Perrault, Mariotte and Halley¶

The decisive test came in France and England in the late 17th century, during the Scientific Revolution described in Chapter 2.

Pierre Perrault (1611–1680), a lawyer and tax official (and brother of Charles Perrault, the author of fairy tales), measured rainfall over three years at Aignay-le-Duc in Burgundy and estimated the area of the upper Seine basin and the annual flow of the river at that point. In De l’origine des fontaines (On the Origin of Springs, 1674), he concluded that the rain falling on the basin was about six times the volume carried away by the river, more than enough to supply it. The rest, he reasoned, was lost to evaporation and used by plants. It was the first quantitative water balance of a river basin.

Edme Mariotte (see Chapter 2) repeated the calculation for the Seine at Paris, with better measurements of the river’s velocity and cross-section, and also measured how rainwater infiltrates into the ground and feeds springs. His Traité du mouvement des eaux (1686) confirmed Perrault’s conclusion.

In England, the astronomer Edmond Halley (1656–1742), better known for his comet and for persuading Newton to publish the Principia, measured the evaporation from a pan of heated water and estimated in 1687 the total evaporation from the Mediterranean Sea. Comparing it with an estimate of the inflow of the rivers flowing into it, he concluded that evaporation from the sea was sufficient to supply the rainfall that feeds the rivers. Together, these studies established the hydrological cycle on a quantitative basis. In 1974 UNESCO and the World Meteorological Organization marked the tercentenary of scientific hydrology, dating it from Perrault’s book.

Evaporation and the atmosphere¶

In 1802 the Manchester chemist John Dalton (1766–1844), better known for his atomic theory, showed that the rate of evaporation from a water surface is proportional to the difference between the vapor pressure at the surface and that of the surrounding air, and that it increases with wind speed. Dalton also made one of the first attempts to calculate the water balance of England and Wales, comparing rainfall with river flow and evaporation.

Dalton’s law of evaporation is the ancestor of all “mass-transfer” methods of estimating evaporation, but it requires the temperature of the surface, which is usually not known. The physics of evaporation would be completed only in the 20th century, by combining it with the energy balance (see below).

Groundwater: from Darcy to Theis¶

Henry Darcy’s experiments on flow through sand columns in Dijon (1856) gave the law of groundwater flow (see Chapter 4):

Jules Dupuit (1863) used Darcy’s law to calculate the flow to wells and the shape of the water table, assuming nearly horizontal flow; Adolf Thiem (1906) applied it to measure aquifer properties from the drawdown around pumping wells. In 1935 Charles V. Theis of the US Geological Survey, using an analogy with the conduction of heat suggested by the mathematician Clarence Lubin, derived the solution for the drawdown around a well pumped at a constant rate in a confined aquifer as a function of time. The Theis equation made it possible to determine the transmissivity and storage of aquifers from pumping tests, and it remains the basis of well hydraulics. In 1940 M. King Hubbert gave a rigorous physical foundation to Darcy’s law and to the concept of hydraulic potential.

Floods: from formulas to the unit hydrograph¶

For engineers designing culverts, bridges, sewers and spillways, the main question was the peak flood flow. The first formulas were empirical.

The rational method. In 1851 the Irish engineer Thomas Mulvaney, working on drainage in Ireland, introduced the idea of the time of concentration, the time for water to flow from the most distant point of a catchment to the outlet, and related the peak discharge to rainfall intensity over that time. In the form $Q = C i A$ (runoff coefficient × rainfall intensity × area), popularized by Emil Kuichling in Rochester, New York (1889), and David Lloyd-Davies in Britain (1906), the rational method is still used for small urban catchments.
Flood frequency. At the beginning of the 20th century, engineers began to treat floods as random events. Weston Fuller (1914) related flood magnitudes to their return periods for American rivers, and Allen Hazen (1914) introduced probability paper. In 1941 Emil Gumbel applied the statistical theory of extreme values to floods, giving the Gumbel distribution and a rigorous definition of the return period.
The unit hydrograph. In 1932 LeRoy K. Sherman proposed that the hydrograph of direct runoff from a catchment produced by one unit of effective rainfall in a given duration has a characteristic shape, and that the response to any rainfall can be obtained by adding up (superposing) scaled and shifted unit hydrographs. This linear, time-invariant view, later written as a convolution integral by James Dooge and others, $Q(t) = \int_0^t u(t-\tau)\, i(\tau)\, d\tau$ , made the unit hydrograph the main tool of flood hydrology for decades. Synthetic unit hydrographs (Snyder 1938; the US Soil Conservation Service in the 1950s) and conceptual models of linear reservoirs (Clark 1945; Nash 1957) followed.

Horton and the physics of runoff¶

Robert E. Horton (1875–1945), an American engineer and one of the founders of modern hydrology, studied infiltration, the process by which rainfall enters the soil. In 1933 he proposed that the infiltration capacity of a soil decreases with time during a storm toward a constant value,

f(t) = f_c + (f_0 - f_c)\, e^{-kt},

(3)

and that surface runoff occurs when rainfall intensity exceeds the infiltration capacity (Hortonian or infiltration-excess overland flow). In 1945 he published a long paper on the erosional development of streams, introducing quantitative laws of drainage networks (stream order, bifurcation ratio) that founded quantitative geomorphology. Horton also helped establish hydrology as a scientific discipline, notably in the Section of Hydrology of the American Geophysical Union, founded in 1930.

The physics of soil water advanced in parallel. W. Heber Green and G. A. Ampt (1911) gave a physically based model of infiltration; Lorenzo A. Richards (1931) combined Darcy’s law with continuity to derive the equation for water flow in unsaturated soil, the Richards equation; John Philip (1957) solved it for infiltration. Later field studies in forested catchments, in the 1960s and 1970s, showed that in humid regions most stormflow is produced not by Hortonian overland flow but by saturation of the soil near streams and by subsurface flow, a reminder that hydrological theories must be tested in the field.

Evaporation from energy: Penman, Thornthwaite and Budyko¶

Evaporation is the largest loss in the water balance of most land areas, and it requires energy: about 2.45 MJ to evaporate one kilogram of water. In 1948 two different approaches appeared.

Charles Warren Thornthwaite (1899–1963), an American climatologist, introduced the concept of potential evapotranspiration (the evaporation that would occur if water were not limited) and estimated it from air temperature and day length, using it to classify climates by their water balance.

In the same year, Howard Penman (1909–1984), a physicist at Rothamsted Experimental Station in England, combined Dalton’s mass-transfer approach with the surface energy balance. By eliminating the unknown surface temperature, he obtained an equation for evaporation from open water using only standard weather measurements: net radiation, air temperature, humidity and wind speed. In 1965 John Monteith extended the equation to vegetated surfaces by introducing a surface (stomatal) resistance:

Charles Priestley and Robert Taylor (1972) simplified the equation for well-watered surfaces, showing that evaporation is roughly 1.26 times the equilibrium evaporation determined by available energy alone.

In the Soviet Union, the climatologist Mikhail Budyko (1920–2001) approached evaporation from the scale of whole climates. In his books on the heat balance of the Earth’s surface (1956) and Climate and Life (1971; English edition 1974), he argued that the long-term mean evaporation from a land area is limited either by the supply of water (precipitation) or by the supply of energy (net radiation), and that the ratio of actual evaporation to precipitation is a function of the ratio of potential evaporation to precipitation, the aridity index. Building on earlier relationships by Schreiber (1904) and Ol’dekop (1911), the Budyko curve became a central framework for understanding catchment water balances and their sensitivity to climate change. Fu Baopu (1981) and Zhang Lu and colleagues (2001), among others, gave one-parameter forms that are widely used today.

A related idea came from the French scientist Robert Bouchet (1963), who proposed that actual and potential evaporation are coupled through the atmosphere: as a land surface dries, the energy not used for evaporation heats and dries the air above it, so potential evaporation rises. This complementary relationship was developed by Morton (1983) and Brutsaert and Stricker (1979), and it allows evaporation to be estimated from routine weather data without knowing soil moisture.

The statistics of long records¶

In the 1950s, the British engineer Harold Edwin Hurst (1880–1978), who spent most of his career studying the Nile for the design of reservoirs, including the Aswan High Dam, analyzed long records of river flows, rainfall, tree rings and lake sediments. In 1951 he found that the range of cumulative departures from the mean grows with the length of record faster than random independent fluctuations would predict, as $n^{H}$ with $H \approx 0.7$ rather than 0.5. The Hurst phenomenon showed that hydrological records contain long-term persistence: droughts and wet periods cluster over decades. Benoit Mandelbrot and James Wallis (1968) interpreted it with fractional processes and named the two kinds of extremes after Biblical stories: the Noah effect (extreme floods) and the Joseph effect (long runs of fat and lean years). Stochastic hydrology, the generation of synthetic streamflow records for reservoir design, grew from this work.

Computers, models and the International Hydrological Decade¶

The computer transformed hydrology just as it transformed fluid mechanics (see Chapter 6). The Stanford Watershed Model (Norman Crawford and Ray Linsley, 1966) represented a catchment as a set of connected storages, simulating the whole land phase of the hydrological cycle continuously in time. It was followed by many conceptual models: the Japanese tank model (Masami Sugawara), the US National Weather Service’s Sacramento model (1973), the Swedish HBV model (Sten Bergström, 1970s) and others still used in operational forecasting.

In 1969 Allan Freeze and Richard Harlan proposed a “blueprint” for a physically based model that would solve the equations of surface, unsaturated and groundwater flow together. The European Hydrological System (SHE, 1986) tried to realize it. In 1979 Keith Beven and Mike Kirkby proposed TOPMODEL, which used topography to predict where saturation and runoff occur. As models multiplied, hydrologists faced the problems of calibration and uncertainty. Beven’s concept of equifinality, that many different parameter sets and model structures can fit the same data equally well, and the GLUE method (Beven and Binley, 1992) for estimating predictive uncertainty challenged the field to be honest about what models can and cannot tell. Vit Klemeš (1986) warned against “dilettantism” in hydrology: complex models calibrated to data, he argued, are not the same as understanding.

International cooperation grew at the same time. The International Hydrological Decade (1965–1974), organized by UNESCO, established networks of representative basins and standardized measurements worldwide; it was followed by the permanent International Hydrological Programme (from 1975). Ven Te Chow’s Handbook of Applied Hydrology (1964) and, later, Applied Hydrology (Chow, Maidment and Mays, 1988) defined the curriculum of engineering hydrology for generations of students.

From catchments to the planet¶

From the 1970s, satellites began to observe the water cycle from space: land cover and snow from Landsat (1972), rainfall from the Tropical Rainfall Measuring Mission (1997), changes in total water storage from the gravity measurements of GRACE (2002), and river and lake levels from SWOT (launched 2022). Land-surface models coupled to climate models brought hydrology into the study of global change.

In 2008 Chris Milly and colleagues argued in Science that “stationarity is dead”: because of climate change and human interventions, the statistics of past floods and droughts can no longer be assumed to represent the future. The International Association of Hydrological Sciences (IAHS) organized two scientific decades to respond to such challenges: Predictions in Ungauged Basins (PUB, 2003–2012), on how to predict streamflow where there are no measurements, and “Panta Rhei — Everything Flows” (2013–2022), on change in hydrology and society. The name chosen for the second decade brings this book full circle to Heraclitus: rivers, catchments, climates and the societies that depend on them are all changing together. A new research field, socio-hydrology (Sivapalan and colleagues, 2012), studies the two-way interaction between people and water, such as how floods shape settlement and how levees change flood risk. The current IAHS decade (2023–2032), HELPING, focuses on solving water problems by engaging local people and knowledge.

Summary¶

Early hydrological records (nilometers, Indian and Korean rain gauges) served agriculture and taxation; Korea’s cheugugi network (1441) and the Seoul rainfall record (1777–1907) are among the most notable.
Perrault (1674), Mariotte (1686) and Halley (1687) established the hydrological cycle quantitatively with the water balance.
Dalton (1802), Darcy (1856) and Theis (1935) founded the physics of evaporation and groundwater flow.
Flood hydrology evolved from the rational method (1851) and frequency analysis (1914–1941) to the unit hydrograph (1932) and Horton’s infiltration theory (1933).
Penman (1948), Monteith (1965), Budyko (1956–1974) and Bouchet (1963) explained evaporation through energy and water limitation.
Computers enabled watershed models from the 1960s; the field now addresses uncertainty, non-stationarity and the coupled change of water and society.