The paper defines the quantity of interest as: "The net water flux at the surface - precipitation minus evapotranspiration over land or precipitation minus evaporation over ocean (P − E) - is a key variable describing the response of the water cycle to climate change." It further explains that P − E represents the *net* surface water flux, which, through the water balance, links to changes in storage (soil moisture, groundwater, snow, etc.) and runoff.

FAO’s soil water balance framework writes the daily soil water balance as: "S(i) = S(i-1) + (P + I) − (ETc + D + R)" where S is soil water content, P is precipitation, I is irrigation, ETc is crop evapotranspiration, D is deep percolation and R is runoff. This formulation shows that the change in soil water storage equals inputs (precipitation and irrigation) minus losses including evapotranspiration and drainage, so wetting or drying is not simply precipitation minus evaporation unless other terms are negligible.

The paper states that the water balance "is often expressed as an equation that relates water inputs, outputs, and storage for a watershed." In its most basic form, the **continuity equation** is written as: **dS/dt = P + Qin − ET − Qout − G**, where S is storage, P is precipitation, ET is evapotranspiration (including evaporation), Q terms are inflows/outflows, and G represents groundwater exchanges. This shows that changes in moisture storage equal precipitation and other inputs minus evaporation/evapotranspiration and other outputs.

The seminar notes define precipitation minus evaporation (P−E) and relate it to moisture transport and storage: "Precipitation minus Evaporation (P-E) • Net supply of Freshwater (land) • Surface salinity and circulation (ocean) • Balanced by moisture transport (atmosphere) & runoff (surface)." A water balance equation is given: "P − E = −∇·F − ΔW" where F is the vertically integrated atmospheric moisture flux and ΔW is the change in atmospheric water content, indicating that P − E is tied to convergence of moisture flux and changes in water storage.

For the land surface, the vertically integrated water balance can be written as: P − E − R = dS/dt, where P is precipitation, E is evapotranspiration, R is runoff, and dS/dt is the change in terrestrial water storage. Over long time scales where dS/dt ≈ 0, precipitation is approximately balanced by evapotranspiration plus runoff.

The continuity equation for terrestrial water storage S can be expressed as: dS/dt = P − ET − R, with P denoting precipitation, ET evapotranspiration and R runoff. Changes in wetness and dryness (terrestrial water storage) are therefore determined by the difference between precipitation and the combined losses through evapotranspiration and runoff.

In its discussion of the global water cycle, the IPCC AR6 notes that the surface water balance links P−E to changes in water storage: it summarizes that over land, "precipitation minus evapotranspiration equals runoff plus change in terrestrial water storage" (expressed in the report’s water balance equations). This formalizes that P−E does not by itself equal the change in moisture but that P−E *minus* runoff (and other terms) gives the change in storage, while globally averaged P approximately equals E over long timescales.

The paper proposes a framework "to quantify the contributions of precipitation and evapotranspiration to soil moisture trends" using a land surface water balance. It explains that soil moisture tendency is governed by the difference between precipitation and evapotranspiration plus terms for runoff and drainage, and uses long-term data to decompose observed soil moisture changes into components attributable to precipitation and evapotranspiration.

Held explains the role of P−E in the atmospheric moisture budget: "So we expect the pattern of precipitation minus evaporation (P-E), which balances the convergence of the net atmospheric water flux, to be enhanced." He notes that globally, "Globally averaged, precipitation balances evaporation (plus transpiration from plants) to an excellent approximation," so that any change in global mean water vapor requires a small sustained imbalance between P and E.

NASA describes how evaporation and precipitation determine net freshwater flux: "Evaporation ("E") controls the loss of fresh water and precipitation ("P") governs most of the gain of fresh water." It states: "Evaporation minus precipitation is usually referred to as the net flux of fresh water or the total fresh water in or out of the oceans." NASA also notes that its goal is to improve measurements of "precipitation (P), evaporation (E), P-E and the land hydrologic state, such as soil-water, freeze/thaw and snow," reflecting that P−E is linked to land moisture conditions and ocean salinity.

Potential evapotranspiration (PE or PET) is defined as "the demand or maximum amount of water that would be evapotranspired if enough water were available (from precipitation and soil moisture)." This concept is used in climate and drought indices that compare precipitation to evaporative demand to characterize wetness or dryness, for example by using the difference between precipitation and potential evapotranspiration.

WHOI describes E−P (and by implication P−E) as the net gain or loss of water by the ocean: "We have compiled modern estimates of evaporation and precipitation over the ocean to develop a more complete picture of water fluxes across the air-sea interface. Taking the difference between evaporation (E) and precipitation (P) tells us the net gain or loss of water by the ocean." It explains that evaporation leaves salt behind and precipitation freshens surface waters, so spatial patterns of E−P affect salinity and thus represent changes in ocean surface freshwater content.

The study "systematically explored soil water evaporation loss at different soil depths" and quantified soil evaporation loss f under continuous evaporation conditions using a nonstationary Craig–Gordon model. It characterizes "the intensity of soil water evaporation relative to local precipitation" by calculating an index (Lc-excess), explicitly relating soil water loss by evaporation to precipitation inputs over time.

The study explicitly links P−E to water resources and moisture conditions: "Changes in precipitation minus evaporation (P–E) are analyzed to investigate the possible impacts of climate change on water resource conditions in China." It highlights that P−E is used as an indicator of how climate change will alter regional water availability, reflecting the balance between incoming (P) and outgoing (E) moisture and its effect on land wetness and dryness.

The article examines "soil moisture evaporative losses in response to wet-dry cycles" and uses mathematical equations recommended by the Food and Agriculture Organization (FAO) for estimating reference evaporation. It discusses how evaporation from soils during wetting and drying cycles removes water from soil storage, linking soil moisture dynamics to the balance between precipitation (wetting events) and evaporation.

This hydrology paper presents the standard terrestrial water balance: "For a given region, the water balance can be written as P − ET − Q = dS/dt, where P is precipitation, ET is evapotranspiration, Q is runoff, and S is terrestrial water storage." The equation shows that the *change in moisture storage* (dS/dt) equals precipitation minus evapotranspiration minus runoff, i.e. P−E alone equals the sum of storage change and runoff, not just storage change by itself.

The fundamental equation for a water balance is: "Change in water storage = Inputs − Outputs" or more formally: dS/dt = I − O, where S is the storage volume of water in the system of interest, I is the rate of water input, and O is the rate of water output. For example: "Runoff = Precipitation − Evapotranspiration ± Storage changes" and, for a soil column, "(P + I + U) − (ET + R + D + L) = ΔS" where P is precipitation and ET is evapotranspiration (evaporation + transpiration).

In describing a model of an exoplanet’s water cycle, the group notes: "The net evaporation field (evaporation minus precipitation) shows that atmospheric water vapor is transported from the night side to the day side." It further states that "Regions on the day side of the planet away from the subsolar point, such as Canada, experience net drying," illustrating that the sign of (E−P) or equivalently negative (P−E) corresponds to drying, while positive P−E would correspond to moistening or net freshwater gain.

The lecture notes state the water balance equation as: "水収支式 P＝E＋D＋ΔS" where P is precipitation, E is evaporation, D is runoff (discharge), and ΔS is the change in storage. It further notes for a catchment that annual evapotranspiration can be approximated as "E = P − Q" (evapotranspiration = annual precipitation − annual runoff), implicitly assuming negligible long-term change in storage.

Discussing the atmospheric water-vapour balance, the paper gives the continuity relation: "なお，水蒸気バランスを考えれば 降水量＝底面からの蒸発量＋g dq/dt dp". Translated: "Considering the water vapour balance, precipitation = evaporation from the lower boundary + g ∫ (dq/dt) dp." This shows that precipitation is related to evaporation plus the rate of change of atmospheric water vapour, not simply equal to evaporation.

In describing the hydrologic (water) cycle, Britannica explains that the water budget of a region is governed by the relationship: precipitation = evapotranspiration + runoff ± storage change. It emphasizes that the water balance involves not only precipitation and evaporation but also runoff and variations in water stored on and below the land surface.

The components of the water-balance equation are **Precipitation (P), Evapotranspiration (ET), Runoff (R), and Change in Storage (ΔS)**. The equation is commonly written as **P = ET + R + ΔS**, expressing that precipitation is balanced by evapotranspiration, runoff, and changes in water storage in the system.

The lesson defines a basic watershed water balance as: **P = Q + E + ΔS + GW**, where P equals precipitation, Q equals runoff, E equals evaporation, ΔS is change in storage, and GW represents groundwater exchange. It explains that the equation is a statement of mass conservation: incoming water from precipitation is balanced by **evaporation/evapotranspiration, runoff, and any change in storage** in soil, surface water, and groundwater.

The abstract states: "A mathematical expression was developed and tested which describes the relation between evapotranspiration and soil moisture." It explains that "a general premise of this mathematical model is that the evapotranspiration-soil moisture relationship is determined by interaction of climatic, soil and plant factors" and presents formulas where actual evapotranspiration ETa is expressed as a function of soil water potential and soil water content, indicating that soil moisture change is controlled by the balance between atmospheric evaporative demand and available water in the soil.

In the accompanying description and lecture, the presenter explains that a field-scale water balance is based on the conservation-of-mass equation: **Change in water storage = Inputs − Outputs**. Inputs include precipitation and irrigation, while outputs include evapotranspiration (evaporation plus plant transpiration), runoff, and drainage. Thus, changes in soil moisture and other storage reflect **precipitation and other inflows minus evaporation/evapotranspiration and other outflows**.

The notes define the water balance equation as: "Input − Output = Change in Storage" and more specifically: **dS/dt = I − O**. One example form is given as **P − R − G − E − T = ΔS or P − R − G − ET = ΔS**, where P = precipitation, R = surface runoff, G = groundwater flow, E = evaporation, T = transpiration, ET = evapotranspiration, and ΔS = change in storage. This shows that the change in water storage (including soil moisture) equals **precipitation minus evapotranspiration and other outputs**.

The water balance equation relates the change in water stored within the basin (S) to its input and outputs: "\( \Delta S=P-ET-Q-D \)." In the equation, the change in water stored within the basin (\(\Delta S\)) is related to precipitation (P) (water going into the basin), and evapotranspiration (ET), streamflow (Q), and groundwater recharge (D) (water leaving the basin). By rearranging the equation, ET can be estimated if values for the other variables are known: \( ET=P-\Delta S-Q-D \).

Water stored in the soil is released into the atmosphere by evaporation and soil moisture is reduced. Evaporation can take place directly from the surface of the soil or through plants (transpiration). In the summer months the combination of lower precipitation and more evaporation makes the soil drier, illustrating that soil moisture changes depend on both precipitation inputs and evaporative losses.

Effective Precipitation (EP) is the amount of precipitation that is actually added and stored in the soil. The moisture deficit is calculated by subtracting the effective precipitation from the calculated evapotranspiration. The equation used in the Farmwest calculator is: "Moisture Deficit wet periods = ETc – Total precipitation", so that a positive deficit indicates that evapotranspiration exceeds precipitation and soil moisture storage is being depleted.

In the soil, daily ET amounts are withdrawals from the account of soil storage. Any rainfall or irrigation is a deposit to soil storage. When an initial soil water value is determined, the water balance can be estimated by subtracting ET daily or over a period of time, showing that changes in soil water storage result from the balance between precipitation/irrigation inputs and evaporative (ET) losses.

In a simple 1D soil column model with no lateral flow and negligible deep percolation over the period considered, the governing equation for volumetric soil water storage S can be written as dS/dt = P − ET, where P is precipitation input and ET is evapotranspiration. Under these idealized assumptions, changes in wetness or dryness are equal to precipitation minus evaporative loss, but in real systems additional terms like runoff, drainage, and capillary rise commonly appear in the full balance.

The lecturer states that in water balance "the inflow to any water system is equal to its outflows plus any changes in storage during the time interval." In a conceptual basin example, precipitation is shown as the **input**, and evaporation or evapotranspiration, runoff, and groundwater outflow are shown as **losses**, with ΔS representing change in water storage. The lecturer summarises this as a balance in which **inflow − outflow ± change in storage = 0**, i.e., change in storage equals inputs minus outputs such as evaporation and runoff.