Hydrology is the backbone of water resources engineering. It deals with the depletion and replenishment of water resources, which is fundamental to the planning, design, and operation of systems like dams, irrigation networks, and storm sewers.

The modern science of hydrology is often traced back to the 17th century with the works of Pierre Perrault, Edme Mariotte, and Edmond Halley. They were the first to quantitatively measure rainfall, runoff, and evaporation to prove that the hydrologic cycle is a closed, quantitative system (e.g., rainfall is sufficient to sustain river flow).

The Hydrologic Cycle (or Water Cycle) is the continuous, solar-driven process by which water is transported from the oceans to the atmosphere, to the land, and back to the oceans. It is a closed system on a global scale, meaning the total amount of water remains constant, though its distribution changes over time and space.

Evaporation is the conversion of liquid water into water vapor from oceans, lakes, and rivers, driven by solar energy. Transpiration is the release of water vapor from plant leaves.

Precipitation is condensed water vapor falling to Earth as rain, snow, hail, etc. Interception occurs when precipitation is caught by vegetation before reaching the ground.

Infiltration is the movement of water through the soil surface into the soil profile. Runoff is water flowing over the land surface to streams and rivers, and eventually to oceans. Groundwater Flow is the movement of water through aquifers (subsurface flow).

Approximately 97% of Earth's water is in the oceans (saline). Of the remaining 3% (freshwater), nearly 69% is locked in glaciers and ice caps, and 30% is groundwater. Only about 0.3% of all freshwater is found in the surface water of lakes, rivers, and swamps.

On a global annual average, precipitation over land is approximately $119,000 \text{ km}^3$, while evaporation from land is about $72,000 \text{ km}^3$. The difference ($47,000 \text{ km}^3$) represents the total annual global runoff to the oceans. Over the oceans, evaporation ($505,000 \text{ km}^3$) exceeds precipitation ($458,000 \text{ km}^3$) by the same amount, balancing the cycle.

Residence time ($T_r$) is the average duration a water molecule spends in a given reservoir before moving to another. For example, deep groundwater can have a residence time of up to 10,000 years, oceans about 2,500 years, lakes around 10 to 100 years, rivers 10 to 20 days, and the atmosphere only about 9 days. This dictates how quickly a water body can flush pollutants.

The water balance equation is based on the Law of Conservation of Mass. For any hydrologic system (e.g., a catchment area or reservoir) over a time interval $\Delta t$, the inflow minus the outflow equals the change in storage.

In this method, outermost tributaries with no branches are designated as 1st-order streams. When two 1st-order streams merge, they form a 2nd-order stream. When two 2nd-order streams merge, they form a 3rd-order stream, and so on. If streams of different orders merge, the resulting stream retains the higher of the two orders. The stream order is a measure of the degree of stream branching within a watershed.

The energy budget dictates how net solar radiation is partitioned at the Earth's surface. It balances incoming net radiation ($R_n$) with sensible heat flux ($H$), latent heat flux ($LE$, which drives evaporation), and ground heat flux ($G$).